Oh, really? Why not? What imposes this limitation? Hilbert's Hotel is surprising and counterintuitive, but I don't see how it rules in or out anything about the nature of the physical universe.
2) Our best theory for the beginning of the universe is the Big Bang, which has a temporal beginning.
Not really. Time itself begins with the Big Bang, the Big Bang does not occur within a flow of time, it starts the flow of time. The question "What was happening before the Big Bang?" is meaningless unless another time dimension apart from the one we experience exists. Positing that sort of idea, however, would merely mean that what we think of as "the universe" isn't really a "universe" at all, because it would then be something less than the sum total of all things, it would merely be a part of a larger phenomena.
Personally, I think the most "rational" decision from a purely objective standpoint is agnosticism.
Atheism and agnosticism are compatible except in the case of "strong" atheism, i.e. adamant insistence that there can't possibly be a God, a position which defines only a small portion of atheists.
We debate what these mean. If you think it makes sense to say, "I don't know," then why are you debating the issue at all?
Because not only do I not know, I think it's highly unlikely that anyone else knows. I don't have to have an alternate answer ready to go in order to say that someone else's answer is probably wrong.
If there is a very large jar of jelly beans, you look at the jar and assert, "The jar contains 12 jelly beans", a quick count of the beans clearly visible can confirm that you are in error as soon as I reach a count of 13, without any knowledge of the actual number of jelly beans, without any more counting being necessary.
If you assert a larger but more likely number given the size of the jar, and tell me you know the answer because an invisible pink unicorn whispered the answer in your ear, I can still assert that you are quite likely wrong, and further assert that even if by chance you are correct your methodology is suspect.
Also, if you're going to call the cosmological argument "circular," I'd like to hear the warrant.
Maybe "self-serving" or "gratuitous" might be better descriptions than "circular", but for me the issue is this: If you start with a contingency argument, that argument is clearly endlessly recursive: every cause requires a previous cause, and there's no beginning that can be reached.
So, how do you break out of that endless loop? One answer that you don't get to break out. The question could simply unanswerable, the assumptions about causality could be incorrect, and the fact that the human mind has a hard time shedding an implicit flow of time from steps in a logical progression could get in the way.
Another answer is that there must be something which is either eternal or self-creating. But you can stop right there. There is no compelling logic that turns that first thing into a God. It doesn't have to be great, it doesn't have to be perfect, it doesn't need a personality, it doesn't need motives.