As others have suggested, your friend is getting it backwards. He's like a hammer asking what a carpenter is useful for.
Given a field (of mathematics, say), there are typically some fields that are more structured than it and others that are less structured. In mathematics, people often say the more structured ones are 'harder', and the less structured are 'softer'. For instance, in increasing order of hardness, we have sets, topological spaces, topological manifolds, differential manifolds, complex manifolds, complex algebraic varieties, algebraic varieties over the rational numbers, integral algebraic varieties. These are in a linear order, but if you throw in other subjects, you'll get a non-linear one. (p-adic algebraic geometry and Riemannian geometry immediately come to mind.)
(I think Gromov has some remarks at the end of an ICM address where he talks about this and gives other examples. Also, don't confuse 'harder' and 'softer' in this sense with what they mean in the sciences, which is essentially 'more precise' and 'less precise'. For instance, in science people say that biology is softer than chemistry. In fact, the two meanings are opposites because in science, more structured objects are less amenable to a precise analysis. But this typically isn't the case in mathematics.)
Now given a subject S and a harder subject H, it's usually true that most objects in S don't admit the structure of an object in H. For instance, most topological manifolds don't admit a complex structure. On the other hand, for the objects of S that do admit such a structure, their theory from the point of view of H is typically much richer than that from the point of view of S. For instance, the study of Riemann surfaces as topological spaces is less rich than their study as complex manifolds. You might say that softer subjects are broad and flexible and harder ones are rich and rigid. Mathematicians tend to view subjects that are softer than their specialty as general nonsense, and harder ones as excessively particular.
This is not to say that a soft field is easier or less interesting than a harder one. Even if it is true that the directly analogous question in the soft subject is easier (e.g. classify Riemann surfaces topologically rather than holomorphically), it just means that the people in the soft subject can move on and study more sophisticated objects. So they just get stuck later rather than sooner. For instance, over the past 50 years, a big fraction of the best number theorists have been studying elliptic curves over number fields. Now elliptic curves over the complex numbers are much easier (I think there hasn't been much new since the 19th century), so the algebraic geometers just moved on to higher genus or higher dimension and are grappling with the issues there, issues that are way out of reach in the presence of arithmetic structure.
Now my main point here is that soft subjects were typically invented to break up the study of harder ones into smaller pieces. (This is surely something of a creation myth, but one with a fair amount of truth.) For instance, the real numbers were invented to break up the study of polynomial equations into two steps: when a polynomial has a real solution and when that real solution is rational. I know very little about modern analysis, but I think that much of it was invented to do the same with differential equations. You first find solutions in some soft sense and then see whether it comes from a solution in the harder sense of original interest.
So the role of soft subjects is to aid in the study of harder ones---people usually don't ask for applications of partial differential equations to the study of topological vector spaces, but it's considered a mark of respectability to ask for the opposite. Similarly, no one talks about applications of engineering to mathematics. Since algebraic geometry is at the hard end of the spectrum above, there aren't many fields in which it is natural to ask for applications. Number theory, or arithmetic algebraic geometry, is harder and of course there are zillions of applications there, but that's not what your friend wants. Just about all mathematicians work in a subject that is softer than some and harder than others (and if you include non-mathematical subjects, then all mathematicians do). That's all good---it takes a whole food chain to make an ecosystem. But it's backwards to ask about the nutritional value of something that typically eats you.
[This picture of mathematics is of course simplistic. There are examples of hard subjects with applications to softer ones. See Donu Arapura's answer, for example. There are also applications of arithmetic algebraic geometry to complex algebraic geometry. For instance, Grothendieck's proof of the Ax-Grothendieck theorem, or the proof of the decomposition theorem for perverse sheaves using the theory of weights and the Weil conjectures. But I think it's fair to say that such applications are the exception---and are prized because of it---rather than the rule.]
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.
Best Answer
I can tell from personal experience that it is possible to learn perfectoid spaces without knowing rigid geometry, just like it is possible to learn schemes or even stacks without knowing much about varieties over complex numbers. In fact, it's even possible to successfully transition to research with this approach. Of course, for the approach to be meaningful/successful you need some "mathematical maturity" (in the sense of being able to clearly distinguish easy/formal parts of the theory from the real meat); for instance, you need a good command of commutative algebra/algebraic geometry (to say the least).
If you want to pursue this, I suggest reading (more or less line by line) Wedhorn's "Adic spaces" for the basics and then (or in parallel) Scholze's "Perfectoid spaces."
Before wise elders start reprimanding me for giving such "irresponsible" advice, let me issue a couple of caveats:
I do agree with the others though that it may be a little bit too late for the AWS to be entirely meaningful. Even if you do not completely understand the lectures, try to isolate the contact points with the material that you've been studying in order to get something out of them.