well to be clear, the theory of evolution by natural selection is our explanation for how life changes over time. so the question of how life originated is not one answered by evolution, per se. so like you said, it would prove that humans evolved from apes (and other creatures before that), but it wouldnt be explanatory as to how the first life came to be.
the working theory there is abiogenesis, but i dont think there is nearly as much evidence on that front as there is for natural selection itself, though its still the most plausible explanation we've got given the evidence that is available (in what we confidently believe the early conditions of the earth were like, we've found that amino acids can develop naturally, which is a big deal)
well its certainly inconsistent with the biblical account of one. but otherwise we'd look to see geological or archeological evidence of a simultaneous global food, which tbh should be pretty apparent given the scale of it... but no convincing evidence on that front either.
i dont think its really the role of science to actively disprove things. rather, the scientific method calls for us to adopt the "null hypothesis"... ie make no assumptions and work from the starting point that a given idea/hypothesis is not the case. the burden then goes on one to make their case in the affirmative. for example, the scientific method for demonstrating the global flood would to start from the assumption that it was not the case, and to only change that position when the evidence for that thing emerges. to reverse that order leads to some absurdities imo, like the flying spaghetti monster stuff (which nobody asserts seriously, but more so to demonstrate how the burden of proof keeps things from falling into absurdity)