etale topology of local schemes

As an addition to Will Sawin's excellent answer: I was wondering about the same thing when I was writing my PhD thesis, and I managed to prove that over $\mathbb{C}$, the etale homotopy type (as defined by Artin and Mazur) of the "algebraic Milnor fiber" (either using henselization or completion) agrees with the homotopy type of the classical Milnor fiber up to profinite completion. In particular, the scheme you mention does not just share the properties of the sphere, it in some precise sense "is" the sphere.

The result can be found in Chapter 4 of my thesis. This is really nothing fancy: it is deduced from the comparison theorems for the fundamental group and cohomology from SGA1 and SGA4 mentioned by Will and a formal argument due to Artin and Mazur, which is probably why I never decided to publish that part. For the comparison of henselization and completion, you need a bit more work, but there are stronger results of this kind in the literature, due to Fujiwara and Gabber.

Let me describe how to answer the etale cohomology questions for a slightly different ring, which is the ring of algebraic elements in the ring of formal power series, also known as the etale local ring / Henselization of the algebraic local ring (although these definitions become different in the greater generality of non-excellent rings). It should be possible to generalize these to the formal power series ring also.

Let $X$ be a variety over $\mathbb C$ and $P$ a point of $X$. Let $R$ be the etale local ring of $X$ at $P$ and $S = \operatorname{Spec} R- P$. We will see that the etale cohomology of $S$, with coefficients in any torsion ring (hence also with $\ell$-adic coefficients), matches the singular cohomology of a punctured topological neighborhood of $X$. So everything known about the singular cohomology transfers to the etale setting, without a new proof. (Although a new proof would be useful to handle the characteristic $p$ case, for instance.)

The etale local ring $R$ of $X$ at $P$ is the forward limit of the rings of functions on all affine etale neighborhoods of $X$ in $R$, so $\operatorname{Spec} R$ is the inverse limit of these neighborhoods, and $S$ is the inverse limit of the punctured neighborhoods. Hence the cohomology of $S$ with coefficients in a torsion ring $\Lambda$ is the inverse limit of the etale cohomology groups of these neighborhoods. This is exactly the stalk cohomology of the sheaf $R j_* \Lambda$ at the point $P$, where $j: X- P \to X$ is the open immersion. This gives an exact sequence $H^*(X,j_!\Lambda) \to H^*(X-p,\Lambda) \to H^*(S,\Lambda)$ where the cohomology of $j_! \Lambda$ is just the cohomology of $X$ relative to $p$.

It follows from Artin's comparison theorem for singular varieties that the first two cohomology groups in the exact sequence are the same as the usual singular cohomology groups. In singular cohomology, this exact sequence is the same as the Mayer-Vietoris exact sequence, and the third term is the singular cohomology of a punctured topological neighborhood.

I can only address your question (1), to which I believe the answer is yes.

Your ring $R$ is a complete intersection local ring of dimension $n$, and hence if $n\ge 3$, then by Stacks 0BPD, it satisfies purity - ie, any finite etale cover of $S = \text{Spec }R - \{m\}$ extends to $\text{Spec }R$. But $R$ is also strictly henselian, so it has trivial etale fundamental group, so the extension of any etale cover of $S$ to $R$ is trivial, so $S$ only has trivial covers, hence $S$ has trivial etale fundamental group if $n\ge 3$.