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.
Consider the group of matrices of the form $w \oplus \det(w)^{-1}$, with $w$ a $4\times4$ permutation matrix. This is an $\operatorname S_4$ inside $\operatorname{SL}_5(\mathbb R) \subseteq \operatorname{SL}_5(\mathbb C)$, and $\mathbb C^5$ decomposes as a sum of the trivial, the sign, and the reflection representation of $\operatorname S_4$. In particular, the space of fixed vectors for $\operatorname S_4$ in $\mathbb C^5$ is $1$-dimensional.
If this $\operatorname S_4$ were contained in an $\operatorname{SO}_3(\mathbb R)$ in $\operatorname{SL}_5(\mathbb R)$, or even in $\operatorname{SL}_5(\mathbb C)$, then the resulting (complex) $5$-dimensional representation of $\operatorname{SO}_3(\mathbb R)$ would be $0^5$, $2 + 0^2$, or $4$.
Since $\sum_{g \in \operatorname{SO}_3(\mathbb Z)} \chi_4(g) = 1(5) + 6(1) + 3(1) + 8(-1) + 6(-1)$ equals $0$, there are no fixed vectors for $\operatorname{SO}_3(\mathbb Z)$ in $4$. Clearly, there is at least a $2$-dimensional space of fixed vectors for $\operatorname{SO}_3(\mathbb Z)$ in $2 + 0^2$ and in $0^5$.
Best Answer
There are no such elements -- the intersection of the derived series of a free group is trivial. In fact, even more is true -- the intersection of the lower central series of a free group is trivial. This is a theorem of Magnus, and by now there are many proofs. The classical one is in the final chapter of Magnus-Karass-Solitar's book on combinatorial group theory.
By the way, a topological proof of this fact (lifting curves to covers to resolve self-intersections, etc) is contained in my paper "On the self-intersections of curves deep in the lower central series of a surface group" with Justin Malestein.
EDIT : I see that you really want finite solvable quotients, not general solvable quotients. It is still true. Fixing a prime $p$, there is a ``mod $p$ lower central series'' of a group whose quotients are $p$-groups (so finite nilpotent if the group is finitely generated). For a free group, Zassenhaus proved in his paper
H. Zassenhaus, Ein Verfahren, jeder endlichen p-Gruppe eine Lie-Ring mit der Charakteristik p zuzuordnen, Abh. Math. Sem. Hamburg Univ. 13 (1939), 200-207.
that the intersection of the mod $p$ lower central series of a free group is trivial. This can also be deduced from the paper I mentioned with Justin Malestein, at least for the prime $2$ (one of the proofs we give actually yields regular covers whose order is a power of $2$).