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.
This is an expansion of my comments above. You do not need resolution of singularities or SGA 4. The key step is "elimination of
ramification" or "Abhyankar's Lemma". This is proved in Append. 1 of Exposé XIII of SGA 1. Here is the link in the Stacks
Project.
Abhankar's Lemma, Stacks Project Tag 0BRM
http://stacks.math.columbia.edu/tag/0BRM
Here is the setup for Abhyankar's Lemma. Let $A$ be a
DVR that contains a characteristic $0$ field, let $A\subset B$ be an
injective, local homomorphism of DVRs with ramification index $e$, i.e.,
$\mathfrak{m}_B^{e+1}\subset \mathfrak{m}_AB \subset \mathfrak{m}_B^e$, let
$K_1/\text{Frac}(A)$ be a finite field extension, let $L_1/\text{Frac}(B)$
be a compositum of $K_1/\text{Frac}(A)$ and
$\text{Frac}(B)/\text{Frac}(A)$, let $A\subset A_1$, resp. $B\subset B_1$,
be the integral closure of $A$ in $K_1$, resp. of $B$ in $L_1$. Let
$\mathfrak{n}_A\subset A_1$ be a maximal ideal that contains $\mathfrak{m}_A
A_1$, and let $\mathfrak{n}_B\subset B_1$ be any maximal ideal that contains
$\mathfrak{m}_B B_1 + \mathfrak{n}_A B_1$.
Abhyankar's Lemma.
if the ramification index $e$ of $A\subset B$ divides the ramification index
of $A\to (A_1)_{\mathfrak{n}_A}$, then $(A_1)_{\mathfrak{n}_A}\subset
(B_1)_{\mathfrak{n}_B}$ is formally smooth, i.e., the ramification index
equals $1$.
Nota bene. In characteristic $0$ this follows easily from the Cohen
Structure Theorem, etc. In positive characteristic, Abhyankar's lemma says
more, because (1) the tensor product $\text{Frac}(B)\otimes_{\text{Frac}(A)}
K_1$ may be nonreduced, and (2) under the assumption that the the residue
field extension $A/\mathfrak{m}_A \to B/\mathfrak{m}_B$ is separable, we
need to also prove that the residue field extension of
$(A_1)_{\mathfrak{n}_A} \to (B_1)_{\mathfrak{n}_B}$ is separable. This
requires a further assumption that $e$ is prime to $p$. When $e$ is not
prime to $p$, Abhyankar's Lemma has counterexamples, but the result that
sometimes does the job is Krasner's Lemma, Stacks Project Tag 0BU9:
http://stacks.math.columbia.edu/tag/0BU9
Let $K$ be a field (not necessarily characteristic $0$), let
$X_K$ be a projective, connected $K$-scheme, and let $x_0\in X_K$ be a $K$-rational point. For every $K$-scheme $T$, an étale cover of $X_T$ trivialized over $x_0$ is a finite, étale morphism $g_T:Z_T\to X_T$ of some degree $d$ together with an ordered $n$-tuple of $T$-morphisms $(s_i:T\to Z_T)_{i=1,\dots,d}$ such that the union of the images of the sections $s_i$ equals $Z_T\times_{X_K} \text{Spec}\kappa(x_0)$.
Rigidity in the Projective Case. For every $K$-scheme $T$, every étale cover of $X_T$ trivialized over $x_0$ is isomorphic to the base change of an étale cover of $X_K$ trivialized over $x_0$, and that trivialized étale cover is unique up to unique isomorphism.
This is, essentially, proved in Section 1 of Exposé X of SGA 1. The key point is rigidity of trivialized étale covers in the proper case.
Descent for Affine Curves.
Now assume further that $K$ is a characteristic $0$ field that contains all roots of unity. Assume that $X_K$ is a smooth, projective, connected curve over $K$. Let
$Y_K\subset X_K$ be a
proper closed subset, i.e., a finite set of closed points. Denote
$X_K\setminus Y_K$ by $U_K$, and assume that the $K$-point $x_0$ is contained in $U_K$.
Fix an integer $d$.
Let
$f_K:\widetilde{X}_K\to X_K$ be a finite surjective morphism of some degree
$n$ with $\widetilde{X}_K$ a smooth, projective curve such that (i)
$f_K^{*}(x_0)$ is a set of $n$ distinct $K$-rational points of
$\widetilde{X}_K,$ and such that for every closed point $y\in Y_K,$ for
every closed point $\widetilde{y}\in \widetilde{X}_K$ with
$f(\widetilde{y})=y$, the ramification index of $\mathcal{O}_{X_K,y}\to
\mathcal{O}_{\widetilde{X}_K,\widetilde{y}}$ is divisible by $e$ for every
$e\leq d$. For instance, begin with a finite morphism $g:X_K\to
\mathbb{P}^1_K$ that is smooth at every point of $Y_K$, such that $g(x_0)$
equals $[1,1]$, and such that $Y_K\subset
f^{-1}(\underline{0}+\underline{\infty})$, and define $\widetilde{X}_K$ to
be the normalization of the fiber product of $g$ and the morphism
$\mathbb{P}^1_K\to \mathbb{P}^1_K$ by $[z_0,z_1]\mapsto
[z_0^{d!},z_1^{d!}]$.
For a field extension $L/K$, the ramification hypothesis on
$f_L:\widetilde{X}_L\to X_L$ over $Y_L$ is still valid. For every finite
surjective morphism $W_L\to X_L$ of degree $d$, for every closed point $w$
of $W_L$ that maps to $Y_L$, the ramification index $e$ at $w$ divides $d!$.
Thus, by Abhyankar's Lemma, the normalization $\widetilde{W}_L$ of
$W_L\times_{X_L} \widetilde{X}_L$ is étale over $\widetilde{X}_L$ at
every closed point lying over $Y_L$. Thus, if $W_L\to X_L$ is assumed to be
étale away from $Y_L$ then $\widetilde{W}_L\to \widetilde{X}_L$ is
everywhere étale of finite degree $d$. Assume further that the fiber of $W_L$ over $x_0$ is a set of $d$ distinct $L$-rational points. Then also the fiber of $\widetilde{W}_L$ is a set of distinct $L$-rational points. By the projective descent result above, there exists a finite, étale morphism $\widetilde{W}_K\to
\widetilde{X}_K$ whose fiber over $x_0$ is a set of distinct $K$-rational points and whose base change equals $\widetilde{W}_L$. Thus, $W_L$ is
an intermediate extension of the base change to $L$ of the extension
$\widetilde{W}_K\to X_K$. In particular, for the fiber $\widetilde{W}_K \times_{X_K} \text{Spec}\kappa(x_0)$ over $x_0$, the degree $n$ morphism $\widetilde{W}_L \to W_L$ defines a partition $\Pi$ of the fiber into $d$ subsets of size $n$.
Descent for $W_L$. There exists a unique irreducible component $W_K$ of the relative Hilbert scheme $\text{Hilb}^n_{\widetilde{W}_K/X_K} \to X_K$ whose fiber over $x_0$ parameterizes the partition $\Pi$ above. The base change of $W_K\to X_K$ to $L$ is isomorphic to $W_L\to X_L$.
In conclusion, for the open
subset $U_K=X_K\setminus Y_K$, for every finite étale morphism
$V_L\to U_L$, there exists a finite étale morphism $V_K\to U_K$ whose
base change to $L$ equals $V_L\to U_L$.
Descent in Arbitrary Dimension.
Now let $k$ be a characteristic $0$ field containing all roots of unity, and let $U_k$ be a normal, quasi-projective variety of dimension $r\geq 1$ together with a specified $k$-rational point $u_0$ in the smooth locus. I claim that there exists a blowing up $\nu_k:U'_k\to U_k$ at finitely many points including $u_0$ such that $U'_k$ is normal, and there exists a flat morphism $\pi:U'_k\to \mathbb{P}^{r-1}_k$ such that the exceptionial divisor $E_0$ over $u_0$ is the image of a rational section of $\pi$. The easiest way to get this is to embed $U_k$ into a projective space $\mathbb{P}^{r+s}_k$, choose a linear $\mathbb{P}^s_k\subset \mathbb{P}^{r+s}_k$ that intersects $U'_k$ transversally in finitely many points including $u_0$, and then let $\pi$ be the restriction to $U_k$ of the linear projection away from $\mathbb{P}^s_k$. Now let $K$ be the fraction field $k(\mathbb{P}^{r-1}_k)$ of $\mathbb{P}^{r-1}_k$, and let $U_K$ be the generic fiber of $\pi$. Let $x_0$ be the $K$-rational point corresponding to the exceptional divisor $E_0$.
For every field extension $\ell/k$, for every finite étale morphism $V_\ell\to U_\ell$ whose fiber over $u_0$ is a set of distinct $\ell$-rational points, the fiber product with $\nu_\ell:U'_\ell\to U_\ell$ is a finite étale morphism to $U'_\ell$ that is trivialized over $E_0$. Thus, setting $L=k(\mathbb{P}^{r-1}_\ell)$, we obtain a finite étale morphism $V_L\to U_L$ whose fiber over $x_0$ is a set of distinct $L$-rational points. Applying the curve case, this descends the generic fiber of $V_\ell\to U_\ell$ to a variety $V_K$ over $K=k(\mathbb{P}^{r-1}_k)$. Finally, we can construct $V_k\to U_k$ as the integral closure of $U_k$ in the fraction field of the $V_K$.
Best Answer
Indeed, you can get whatever genus you want even with a fixed Galois group G, so long as its order is divisible by p: this is a result of Pries: .pdf here.
In fact, Pries has lots of papers about exactly what can happen; looking at her papers and the ones cited therein should give you a pretty thorough picture.
We don't know the Galois group of the affine line, but we do know which finite groups occur as its quotients; this is a result of Harbater from 1994 ("Abhyankar's conjecture for Galois groups over curves.") Update: As a commenter pointed out, Harbater proved this fact for an arbitrary affine curve; the statement for the affine line was an earlier theorem of Raynaud.