This is really just an elaboration of Emerton's comment: You should read Mark Kisins' review of Faltings's paper "Almost etale extensions".
But I wanted to elaborate: Faltings regards the almost purity theorem as an analogue of Zariski-Nagata purity. In Faltings's original setup, it was formulated as follows. Consider the rings
$$ R_m = \mathbb{Z}_p[p^{1/p^m},T_1^{\pm 1/p^m},...,T_n^{\pm 1/p^m}]
$$
Each of them is smooth over $\mathbb{Z}_p[p^{1/p^m}]$ (in particular regular), and the transition maps are finite and etale after inverting $p$. Let $S_0$ be a finite normal $R_0$-algebra, which is etale after inverting $p$, and let $S_m$ be the normalization of $S_0\otimes_{R_0} R_m$. The idea is that there is ramification of $S_0$ in the special fibre, and you want to get rid of it, by adjoining the chosen tower of highly ramified rings $R_m/R_0$. It is not hard to see that this actually works almost at the generic point of the special fibre: At the generic point, the local ring is a discrete valuation ring, and the statement is that the discriminant of the extension of discrete valuation rings becomes arbitrarily small as $m\to\infty$ (this boils down to some more or less classical ramification theory).
Now, assume that you were lucky, and for some $m$, the ramification at the generic point of the special fibre is not just very close to zero, but actually zero on the nose. Zariski-Nagata purity tells you that the ramification locus of $S_m$ over $R_m$ has to be pure of codimension $1$, but you know that there is no ramification at the codimension $1$ points (which are either in characteristic $0$, or equal to the generic point of the special fibre) -- thus, there is no ramification at all, and $S_m$ over $R_m$ is etale.
The almost purity theorem says that this result extends to the almost world: In the limit as $m\to \infty$, $S_\infty$ over $R_\infty$ is almost etale (in the technical sense defined in almost ring theory).
For local noetherian rings, the henselization has much more direct algebro-geometric meaning than the completion since it is built from local-etale algebras. Hence, your statement that the completion provides more intuition than the henselization is just a matter of having less experience with henselizations. See Example 2 below for why there is no need in practice for anything like a "formal topology", which likely doesn't exist anyway. In particular, the answer #1 is likely "no", so #2 is then moot.
But the real reason this is all somewhat of a wild goose chase is that you are correct about #4: for excellent local noetherian rings, the deep Artin-Popescu approximation is the ultimate answer to nearly all questions about passing between their completions and their henselizations. This theorem implies that for any such ring $R$ and any finite system of polynomial equations in several variables over $R$, any solution in the completion can be approximated arbitrarily closely (for the max-adic topology) by a solution in $R^{\rm{h}}$. (The real content, upon varying the system of equations, is that there is even a single solution in $R^{\rm{h}}$, let alone one that is "close" to the given solution in the completion.) However, that raw statement about systems of equations does not do justice to the significance, so let's illustrate with a couple of applications; see section 3 of Artin's IHES paper (http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1969__36_/PMIHES_1969__36__23_0/PMIHES_1969__36__23_0.pdf) for many more striking applications.
Example 1. A pretty algebraic application of this approximation theorem is that if $R$ is an excellent local normal noetherian domain then $R^{\rm{h}}$ is the "algebraic closure" of $R$ in the normal local noetherian completion $\widehat{R}$. Even for $R$ the local ring at the origin of an affine space over a field, I am not aware of a proof of this application without invoking Artin approximation.
Example 2. A geometric reason that Artin-Popescu approximation is so relevant is that it implies the following extremely useful fact: if $R$ is an exellent local noetherian ring and $A, B$ are two local $R$-algebras (with local structure map) essentially of finite type such that there exists an isomorphism $f:\widehat{A} \simeq \widehat{B}$ as $\widehat{R}$-algebras then $A$ and $B$ admit a common residually trivial local-etale neighborhood.
(In particular, $A^{\rm{h}} \simeq B^{\rm{h}}$ over $R$; we definitely do not claim that such an isomorphism of henselizations can be found which induces $f$ on completions.)
Even for $R$ a field, this is not at all obvious but is super-useful; e.g., it comes up when rigorously justifying the sufficiency of proving some facts about the geometry of specific singularities by studying analogous special cases with completions, such as in deJong's paper on alternations or various papers on semistable curves, etc.
It remains to address #3, under reasonable hypotheses on $R$. We will see that the first question in #3 is quite easy, and the second generally has a negative answer when $\dim R > 1$ (whereas the case $\dim R = 1$ is affirmative under reasonable assumptions on $R$, a fact used all the time in number theory; more on this below).
Now we come to the hypotheses that should have been made on $R$ for posing any questions of the sort being asked.
Presumably you meant for $R$ to be noetherian (max-adic completion would not be appropriate otherwise), and also excellent (or else the completion could fail to be reduced when $R$ is reduced, etc.). Moreover, let's forget about strict henselization and only consider henselization, since the latter has the same completion and is the correct "algebraic" substitute for the completion. And since henselization of an excellent local ring is excellent (see the Remark after Prop. 1.21 in the book "Etale Cohomology and the Weil Conjectures" for a proof, which was overlooked in EGA even though preservation under strict henselization is done in EGA), we may as well assume $R$ is henselian.
Since the category of finite etale algebras over a henselian local ring is naturally equivalent to that over its residue field, the invariance of etale fundamental group under passage to the completion for any henselian local noetherian ring is obvious. That is, the first question in #3 has an easy affirmative answer.
I do not know what is the precise intended meaning of the 2nd to last paragraph in the posting (since equicharacteristic 0 is a poor guide to the rich arithmetic structure of mixed characteristic $(0,p)$), but one might contemplate trying to use the Artin-Popescu approximation theorem to affirmatively settle:
Question. Let $R$ be an excellent henselian local domain that is normal, with fraction field $K$. Let $\widehat{K}$ denote the fraction field of the completion $\widehat{R}$. Do the Galois groups of $K$ and $\widehat{K}$ naturally coincide?
Despite whatever intuition may be derived from Artin-Popescu approximation, the possibly surprising answer to the Question is generally negative when $\dim R > 1$, even when $R$ is regular. (I only realized that after failing to prove an affirmative answer beyond dimension 1 in this way.)
Before discussing counterexamples, a few Remarks will be helpful to put things in context:
Remark 1: That the local noetherian ring $\widehat{R}$ is actually a normal domain requires a proof. One way to see it is to use Serre's homological characterization of normality and the fact that the flat map ${\rm{Spec}}(\widehat{R}) \rightarrow {\rm{Spec}}(R)$ has regular fibers (since $R$ is excellent); see the Corollary to 23.9 in Matsumura's "Commutative Ring Theory". Note also that when $\dim R > 1$, there is no "topological field" structure on $K$ and so $\widehat{K}$ is not a "completion" for $K$ in some intrinsic sense not referring to $R$; it is just notation for the fraction field of the completion of $R$.
Remark 2. In case $\dim R = 1$ the Question has an affirmative answer (even valid for rank-1 henselian valuation rings and valuation-theoretic completions thereof, without noetherian hypotheses; see 2.4.1--2.4.3 in Berkovich's paper in Publ. IHES 78). This does not require the approximation theorem, and is an instructive exercise in the use of Krasner's Lemma. The main task when $\dim R > 1$ is to try to replace Krasner's Lemma with an argument using the approximation theorem, bypassing the fact that $K$ (let alone its finite separable extensions) doesn't have a natural topological field structure when $\dim R > 1$; all such attempts are doomed to fail, as we'll see below.
Remark 3. Equality of the Galois groups (through the evident natural map) would imply via Galois theory that every finite separable extension of $\widehat{K}$ arises by "completing" a finite separable extension of $K$ and more generally that $E \rightsquigarrow E \otimes_K \widehat{K}$ is an equivalence of categories of finite etale algebraic over $K$ and $\widehat{K}$ respectively. This equivalence is very useful in number theory for the case $\dim R = 1$ when the answer is affirmative.
To see what goes wrong beyond dimension 1, consider a general $R$ as above, allowing $\dim R$ arbitrary for now and not yet assuming regularity. Let $K'/K$ be a finite separable extension, so by normality of $R$ the integral closure $R'$ of $R$ in $K'$ is module-finite over $R$. Since $R$ is henselian and $R'$ is a domain, it follows that $R'$ must be local (and henselian and excellent). Thus, $R' \otimes_R \widehat{R} = \widehat{R'}$ is a normal domain, and its fraction field $\widehat{K'}$ is clearly finite separable over $\widehat{K}$ with degree $[K':K]$.
In case $K'/K$ is Galois with Galois group $G$, it follows that $G \subset {\rm{Aut}}(\widehat{K'}/\widehat{K})$ with $\#G = [\widehat{K'}:\widehat{K}]$, so $\widehat{K'}/\widehat{K}$ is Galois with Galois group $G$. In this way, we see (by exhausting a separable algebraic extension by finite separable subextensions) that $K_s \otimes_K \widehat{K}$ is a Galois extension of $\widehat{K}$ with Galois group ${\rm{Gal}}(K_s/K)$. The Question is exactly asking if this is a separable closure of $\widehat{K}$.
By Galois theory over $\widehat{K}$, it is the same to ask that any given finite Galois extension $E/\widehat{K}$ is the "completion" (in the above sense based on integral closures of $R$) of some finite separable extension $K'/K$, necessarily of the same degree.
Now let's see that already for quadratic extensions one has counterexamples with $R$ any countable henselian excellent regular local ring with $\dim R > 1$ and ${\rm{char}}(K) \ne 2$, such as the henselization at any closed point on a smooth connected scheme of dimension $>1$ over a field not of characteristic 2; e.g., $R = \mathbf{Q}[x_1,\dots,x_n]_{(x_1,\dots,x_n)}^{\rm{h}}$ is a counterexample for any $n > 1$.
It is the same to show for such $R$ that the natural map $K^{\times}/(K^{\times})^2 \rightarrow \widehat{K}^{\times}/(\widehat{K}^{\times})^2$ is not surjective. Since $K^{\times}$ is countable, it suffices to show that the target is uncountable. This uncountability feels "obvious" when $\dim R > 1$, but we want to give a rigorous proof (e.g., to understand exactly how the condition $\dim R > 1$ and the completeness of $\widehat{R}$ are used).
The regularity of $R$ implies that of $\widehat{R}$, so $\widehat{R}$ is a UFD. Thus, the height-1 primes of $\widehat{R}$ are principal, corresponding to irreducibles in $\widehat{R}$ up to unit multiple. Any two irreducibles that are not $\widehat{R}^{\times}$-multiples of each other give rise to distinct elements in $\widehat{K}^{\times}/(\widehat{K}^{\times})^2$, by the UFD property. Thus, it suffices to show that $\widehat{R}$ contains uncountably many height-1 primes.
Complete local noetherian rings satisfy "countable prime avoidance": if an ideal $I$ is contained in a union of countably many prime ideals then it is contained in one of those primes. (This fact is originally a lemma of L. Burch. It can also be deduced easily from the Baire category theorem by using that such rings with their max-adic topology are complete metric spaces in which all ideals are closed, as noted in section 2 of the paper "Baire's category theorem and prime avoidance in complete local rings" by Sharp and Vamos in Arch. Math. 44 (1985), pp. 243-248.) By the UFD property, every non-unit in $\widehat{R}$ is divisible by an irreducible element and so lies in a height-1 prime. Thus, the maximal ideal $m$ of $\widehat{R}$ is contained in the union of all height-1 prime ideals. But $m$ is not contained in (equivalently, not equal to!) any of those height-1 primes since $\dim \widehat{R} = \dim R > 1$ (!), so by countable prime avoidance in $\widehat{R}$ it follows that $\widehat{R}$ must have uncountably many height-1 primes, as desired. (Related reasoning comes up in Example 2.4 in the paper of Sharp and Vamos mentioned above.)
Remark: For any $R$ as in the above counterexamples, and $B'$ the module-finite integral closure of $\widehat{R}$ in a quadratic extension of $\widehat{K}$ not arising from a quadratic extension of $K$, there is no finite $R$-algebra $B$ satisfying $B \otimes_R \widehat{R} \simeq B'$. This may seem to contradict Theorem 3.11 in Artin's IHES paper linked above, but it does not (since there is no reason that the proper closed non-etale locus in the base for the generically etale ${\rm{Spec}}(B') \rightarrow {\rm{Spec}}(\widehat{R})$ is contained in a proper closed subset of ${\rm{Spec}}(\widehat{R})$ that is the preimage of one in ${\rm{Spec}}(R)$).
Best Answer
Lol @"varying degrees of enthusiasm" ;-). And sorry for the late answer...
Let me try to answer your questions. First, for any connected analytic adic space $X$, say, with a geometric point $\overline{x}\to X$, one can define $\pi_1^{\mathrm{et}}(X,\overline{x})$ just like in SGA1 for schemes, by looking at the Galois category of finite etale covers of $X$. In particular, passing to an inverse limit of all such finite etale covers equipped with a lift of $\overline{x}$, one can define a (profinite) "universal cover" $\tilde{X}\to X$. If $X$ lives over $\mathbb Z_p$ and is affinoid (probably Stein is enough) then $\tilde{X}$ is perfectoid; see for example Lemma 15.3 here (the funny phrasing there is solely due to the desire to also handle the case that $X$ is not connected).
This largely answers question 2). Unfortunately, I don't know how to define a pro-etale fundamental group in the spirit of my paper with Bhatt. There we handle the case of schemes that locally have only a finite number of irreducible components. This is a very mild condition for schemes, but for analytic adic spaces, the condition is much too strong, see Example 7.3.12 of our paper. That example shows that the formalism actually does not work in the same way for analytic adic spaces, and I don't know how to correct it. So I will only use the usual $\pi_1^{\mathrm{et}}$.
For question 1), the answer is actually No. Using Artin-Schreier covers, there are lots and lots of finite etale covers beyond the ones one might think about, so in particular the perfectoid closed unit disc has very large $\pi_1^{\mathrm{et}}$ (even (or especially) pro-$p$). What one might reasonably hope is that any finite etale cover of degree $p$ of the punctured perfectoid closed unit disc extends to a finite etale cover of the perfectoid closed unit disc. For this precise question, I'm actually confused: If the finite etale covers comes from some finite stage, it follows from some classical results in rigid geometry that it extends to a finite, possibly ramified cover, over the puncture, and then by Abhyankar's lemma this becomes trivial after passing to the perfectoid cover. However, I believe that at infinite level, one will get new, more nasty covers, that do not come from finite level.
About question 3): One key fact is that affinoid perfectoid spaces have etale $p$-cohomological dimension $\leq 1$, i.e. for etale $p$-torsion sheaves, etale cohomology sits in degrees $\leq 1$. This in fact reduces by tilting to the case of characteristic $p$, where it follows from Artin-Schreier theory. Combining this with some interesting examples of perfectoid towers, one can get interesting vanishing results. In fact, these can usually be slightly improved upon by using $\mathcal O_X^+$-cohomology, the primitive comparison theorem, and the (almost) vanishing of $\mathcal O_X^+$-cohomology on affinoid perfectoids. This has been applied for example to Shimura varieties, abelian varieties [Well, the written version of that paper actually doesn't use this method, but our original approach did use it, see the discussion on page 1], and moduli spaces of curves.
This ties in with question 4). What one usually does is the following. Say $\ldots\to X_2\to X_1\to X_0$ is some tower of proper rigid-analytic varieties over $\mathbb C_p$ with perfectoid limit $X_\infty$. For each $X_n$, the primitive comparison theorem says that $$H^i(X_n,\mathbb F_p)\otimes \mathcal O_{\mathbb C_p}/p\to H^i(X_n,\mathcal O_{X_n}^+/p)$$ is an almost isomorphism, where both sides are etale cohomology. (The proof of this uses some Artin-Schreier theory, and one could also formulate an Artin-Schreier sequence, but this tends to give weaker results.) Passing to the colimit over $n$ (so the limit on spaces), one sees that also $$H^i(X_\infty,\mathbb F_p)\otimes \mathcal O_{\mathbb C_p}/p\to H^i(X_\infty,\mathcal O_{X_\infty}^+/p)$$ is an isomorphism. Now on perfectoid $X_\infty$, the group on the right behaves like coherent cohomology, in particular it can (almost) be computed on the analytic side, and in fact by a Cech complex. This shows in particular that it (almost) vanishes in degrees larger than $\dim X_\infty$. In particular, $H^i(X_\infty,\mathbb F_p)$ vanishes in degrees larger than $\dim X_\infty$, which gives the vanishing theorems I mentioned.