Thanks for the answer-- I accepted the other one just because it addressed both questions, but this is a helpful perspective

Interesting-- thank you for the details

It’s worth noting that Morita equivalence (in the sense of two Azumaya algebras having equivalent categories of right modules) is the same as the equivalence relation described in the original post when working over an affine scheme, but in general the former is coarser than the latter. See example 1.3.16 here:people.math.wisc.edu/~caldararu/publications/…

I don't see why the Weierstrass factorization theorem can't apply in a canonical way. You can just choose an ordering on the zero set in order of nondecreasing modulus, and choose your elementary factors $E_{p_n}$ with $p_n = n$. See Conway, Functions of One Complex Variable, theorems VII.5.12 and VII.5.15. For whatever natural complex structure you choose on the spaces involved I expect this section would be analytic.

For a general topological space, it's probable that every property of $X$ which is determined by the fundamental group structure can be phrased as a property of $X$'s algebraic invariants. For a manifold or CW complex or something with additional structure, you can get more specific about "geometric" properties. Ultimately what the fundamental group controls is covers, and these provide a more geometric interpretation than the definition via loops.

If $X$ is just a general topological space, it's not clear what you mean by "geometric" property and whether it's distinct from a topological property. The fundamental group affects other algebraic invariants of $X$, such as the first homology group, or via the Hurewicz or Freudenthal suspension theorems. The fundamental group determines a compact connected surface up to isomorphism. There is also the Mostow rigidity theorem which says a certain class of manifolds are totally determined by their fundamental group.

I see— this answer is incomplete then. Thanks for the info

Smaller question: what can we say when $N$ is a single point?

The pushforward is always defined as a sheaf. Are you asking for a criterion for it to be a vector bundle?

Starting with a local system, can you define the map as follows? Take your fiber $V$ over a basepoint $x \in X$, pull it back via a local isomorphism $p$ to a point $\tilde{x} \in \tilde{X}$ over $x$, apply a deck transformation $g$, and then define the representation to act on $V$ via $\rho(g) = \pi_* \circ g \circ (p_*)^{-1}$. If you have a reasonably explicit description of your covering map I think you can make this pretty explicit. Perhaps you run into a similar problem by trying to explicitly describe the pullback of your local system to a local system on $\tilde{X}$ though

Interesting question-- unless I am missing something, it's worth noting that any variety over any field is (the base change of) a finite type scheme over the integers (it's quasi-compact and affine-locally we can define it via a finite number of equations whose coefficients we can take to be in a finitely generated field). So you can ask for a reformulation which specializes to the full original statement of the Tate conjecture, right?