In order for Premise 2 of the Leibnizian Cosmological argument to hold—the premise that requires an agent with will to bring the universe into existence—does one have to postulate that the Big Bang is an event in time rather than the explosion of time and space into existence?
I would say that the Big Bang is the first event in time. God creates the universe at moment of time T=0, or, stated differently, the act of creating is simultaneous with the universe beginning to exist. So the moment of creation is the first event in time. If the universe is genuinely temporal, it would require a first moment-- a beginning. Things which begin are caused. Eternal causes without wills (like a timeless vacuum as in a vacuum fluctuation model) necessarily have eternal effects. Only a timeless being with a will can account for a universe with a beginning.
Hawking admits that on real time models, there is in fact a beginning, and that this could suggest a Beginner: “So long as the universe had a beginning, we could suppose it had a creator” (A Brief History of Time, p. 140-141). This is why his quantum gravity model is so important-- it curves time to get around the beginning point. Of course, numerous physicists, including Roger Penrose of the Penrose-Hawking Singularity Theorems, do not find Hawking's system of explaining the universe to be correct or even, in the case of Penrose, scientific.
How can you reconcile the omnipotent nature of god—which would require that he be able to end his own existence if he so chose—and traits god possesses like emotion and thought—which are contingent upon neurology and evolution in every instance we can examine—with your claim that god exists necessarily, and therefore must not be able to cease his own existence or possess traits that are contingent?
I think this problem stems from you defining omnipotence in a way which Christian theologians do not. I take omnipotence to mean that God can do anything at all which does not deny His nature. (such as things which are irrational, immoral, etc.) For instance, because God is the ground of rationality, and God is consistent with Himself, He will not deny Himself by behaving in irrational ways. For example, God could not draw a square circle or make a rock so heavy that He couldn't lift it. Of course, these are anthropomorphisms which make God material, but even with this flaw aside, God could not do these things because they are irrational. If God cannot deny Himself by behaving irrationally, but can still be called omnipotent in a meaningful sense by being able to do anything which is right and rational to do, it follows that He also cannot cease existing because to do so would be an act of denying His nature and doing that which is not rationally possible.