What would lead someone to the conclusion that God did NOT create us and the world around us?
Depends on the god in question, but there are potentially many reasons. For instance, if the God you're talking about is a good god, then the problem of evil directly suggests that this god did not create the universe.
Any time that a particular conception of God (e.g. that God is good) creates a prediction about the universe (e.g. that the universe will have no more evil and suffering than strictly necessary), we can test that prediction against the actual universe and see if it's correct.
Also, we have quite a bit of knowledge about the development of religion and god-belief that shows how religious creation beliefs arise. If we have good evidence that they're a human product of innate psychology and cultural development (and I think we do), then we already have the explanation: people believe in God because our brains are predisposed to this sort of belief and we're generally brought up to believe certain things about certain gods. And if this is the actual cause of religious beliefs, then we can conclude that other potential explanations (e.g. the beliefs are actually true) aren't the cause. This doesn't completely exclude the possibility of God, but it does render such a god irrelevant as an explanation for the physical universe.
Why on earth would the scenario of a random existance, and this fantastically complex world being some sort of accidental happening, be any more believable than the concept of a Creator? Both are theories and are unprovable positions.
Well, every bit of physical evidence we have is consistent with the things we see being created by unguided processes. I agree that this doesn't necessarily meant that some god or gods isn't lurking out in the shadows somewhere, but it does imply that such a god, if he/she/it/they exist, is insignificant in terms of effect on the universe.
Also, since God doesn't actually provide any explanatory power but creates many new questions that demand answers themselves, I think that we're left in a situation where, in the absence of evidence one way or the other, a universe without God is always more plausible than a universe with God.