>The universe exists. Is it eternal or did it have a beginning?
Our universe appears to have had a beginning, namely at the point of the Big Bang. However, it is highly probable that there exists an Omniverse of which our universe is only a part, and that this Omniverse is eternal (although it's conceivable that it might not be eternal). Alternatively, 'eternal' in the sense of our perception of time may not be the same thing as 'eternal' in the sense of absolute time. So there are a lot of possibilities to take into account.
>It could not be eternal since that would mean that an infinite amount of time had to be crossed to get to the present. But, you cannot cross an infinite amount of time
Ahem. That is incorrect. It is not possible for a sentient being (like a human) to EXPERIENCE an infinite amount of time, however there is nothing stating that the experience of a sentient being cannot start at a point in infinite time.
As an analogy, imagine an infinitely long line with dots on it spaced every two centimeters. The line has no ends, it extends forever in both directions. Imagine that you start at a dot on this line and count ten dots forwards. Is there anything impossible about doing this? No. It IS impossible to count to the point in front of you from the beginning of the line, but it is NOT impossible to count ten dots starting at the point in front of you. And time is the same way: You can't pass an infinite amount of it, but there's certainly nothing preventing you from starting in the middle of it and passing a finite amount from there.
>Therefore, the universe had a beginning.
As I have demonstrated above, that is not a necessary conclusion. But for the moment I'll humor you and imagine that it is.
>Something cannot bring itself into existence.
Not as far as we can tell, no. Our universe runs on a kind of logic that doesn't seem to allow things to arise from nothing. However, it is conceivable that the Omniverse above us uses a higher kind of logic by which it IS possible for something to arise from nothing, in which case there is really nothing preventing it from simply popping into existence. Alternatively, it's not impossible that at some level the Omniverse is not even logical at all. Personally I think both of these are rather unlikely (and the latter less likely than the former), but they are still possible.
>Therefore, something brought it into existence.
But you've already stated that one, nothing can be eternal, and two, something cannot arise from nothing. If these are stated as absolutes, then this 'something' you're talking about follows the same rules, which means it must in turn have a cause, which must also have had a cause, which must ALSO have had a cause, and so on ad infinitum. What you are doing here is stating some rules, and then using those rules to demonstrate the existence of something which doesn't obey the rules. But don't you understand that this is self-defeating? If the thing doesn't obey the rules, then the rules must have never really applied in the first place, which in turn means it is no longer necessary for the thing to exist.
In fact, the only thing you HAVE managed to prove through all this is that one or the other of your initial rules must be false. If everything must have a cause (your second rule), then the string of causes goes back forever and is therefore eternal (violating your first rule). On the other hand, if there is no eternity (your first rule), then there must have been a moment before which nothing existed, which means that something must have arisen from nothing (violating your second rule). Your two rules are mutually contradictory, and therefore you can't validly draw any conclusions from assuming them both to be true.
>It would have to be greater than the universe and be a sufficient cause to it.
No, it would not actually have to be greater than the Universe in the sense of informational complexity (which is the most basic measure of size). It has been shown by experiments that apparently complex information can arise from simple information. In fact, a whole book has been written about this, you can read it here:
http://www.wolframscience.com/nksonline/page-1?firstview=1
>The Bible promotes this sufficient cause as God.
Have you ever heard of the phrase 'jumping to conclusions'? Because that's what you're doing here. You go on for a while about the basic idea that the Universe has a first cause, and then from there you go straight to the christian God without an ounce of logic inbetween. It's not like there aren't hundreds, even thousands of other gods from various religions around the world which could also fill this position of a 'first cause'. Why do you immediately jump to the christian God rather than any other? If you think the christian God is the only one that explains the way the Universe is and that no other gods can, then at LEAST post the trian of reasoning that leads you to that conclusion. Anything else just looks biased, which in turn makes you look a little hypocritical given your attempt to use solid logic right up until this point.
>All things that came into existence were caused to exist. You cannot have an infinite regression of causes (otherwise an infinity of time has been crossed which is impossible because an infinity cannot be crossed). Therefore, logically, there must be a single uncaused cause that did not come into existence.
But if it did not come into existence, then it must also have passed an infinite amount of time. Either there are infinite causes, in which case infinite time has passed, or there is a first cause which did not come from nothing, in which case infinite time has passed, or there is a first cause which has not been around infinitely long, in which case it came from nothing. These are the only three possibilities, and all of them violate at least one of the rules you stated.
>God is the single uncaused cause.
Again, that is just jumping to conclusions. You need a lot more reasoning in order to get from a first cause existing to the christian God existing without looking biased. Most specifically, you need a way to prove that the first cause is necessarily the christian God. You have not yet provided this.