Proof is for mathematics. Arguably (and, I think, correctly), I can prove that I (as self-awareness) exist the way Descartes did, but beyond this everything relies on certain assumptions and arguably even this does. Proofs require formal, closed systems whereby granted axioms and rules allow statements to be necessarily true.
But the "proof" (i.e., the scientific equivalent) is that the universe is expanding and that it makes up all of space (and spacetime). There is no space outside of it, there isn't anything for it to expand into.
Because most multiverse theories don't actually posit multiverses, for one. Rather, the initial big bang expanded in such a way as to allow for "pocket" regions to emerge which differ significantly enough from our own observable region for some physicists to call them "universes" (in a more speculative version, inflationary processes turn these "pockets" into bubble universes which are even more distinct, but not past-time eternal either). Alternatively, the multiverse results from the apparent collapse of the wave function when really each collapse is a new "branch" (universe) in which one of the possible outcomes is observed. However, before the big bang, there were no quantum systems, ergo no eternal multiverse.
Finally, given the most speculative multiverse cosmologies, in which there is an eternal process of bangs and crunches, eternity itself becomes meaningless as time is contingent upon the existence of things like entropy and spacetime.
I wouldn't. In a multiverse in which there are truly causally separate universes, it makes no sense to speak of what the expand into any more than it does our own universe.
The center of the cosmos is the center of the universe. Also, I don't think getting into manifolds, Riemannian geometry, and "shapes" in higher dimensional (non-Euclidean) spaces is particularly fruitful here.