I am representing the opposing (atheistic) position, and will respond to punkforchrist's points on the Thomistic argument for a creator.
The word "contingent", I use here to mean "dependent on external elements".
"Prime Mover" is the same as the "Unmoved Mover".
As I respond to your premise of:
1." Everything that exists has an explanation of its existence, either in the necessity of its own nature or in an external cause."
I can accept that all the things that physically exist are, in fact, contingent beings. My cat, for example, does not just exist. She is a creature dependent on other things, such as food, water, and air. The food chain, indeed, shows that animals are all dependent on one another, as well as plants, which are in turn dependent on air, water and sunlight. All physical beings, can thus be called contingent and have a cause.
With regards to:
2."If things in motion have an explanation for their existence, that explanation is an Unmoved Mover."
I would say that this is contradictory to the premise (1), for surely if all things require a mover, then this should apply to the Prime Mover as well? Infinite regress may, or may not, require an external cause. This is not falsifiable, as we have no way to prove that there was a cause for infinite regress. Furthermore, even if there is an Unmoved Mover, there is no proof that it was the God of Abraham. It could as well have been Odin, Vili and Ve--or even Lucifer.
I accept premise (3), however, I fail to understand the jump in logic to the conclusion of (4), as we have no way of actually verifying this, particularly with the implication that this contains a proof for any specific deity.
Hume's objection--upon which I base my own objection--is based on observable things, such as numbers. You can write what you think is the largest number in existence, and I can then add 1 to it. Equally, I can write what I think is the smallest number in existence, and you can subtract 1 from it. The sequence of those events, dare I say it, is eternal. This quote sums up rather nicely the objection to even suggesting a Prime Mover:
"In such a chain too, or succession of objects, each part is caused by that which preceded it, and causes which succeed it. Where then is the difficulty? But the whole, you say, wants a cause. I answer that the uniting of these parts into a whole, like the uniting of several distinct counties into one kingdom, or several distinct members into one body, is performed merely by an arbitrary act of the mind, and has no influence on the nature of things. Did I show you the particular causes of each individual in a collection of twenty particles of matter, I should think it very unreasonable should you afterwards ask me what was the cause of the whole twenty. This is sufficiently explained in explaining the cause of the parts."
As I have said, the First Cause, even if I should accept your argument, need not be God, but may perhaps be the universe itself. If you say "God is necessary", then it is possible that "the universe is necessary".
Response to the W-PSR:
1.There possibly exists an Unmoved Mover.
For the sake of discussion, I will accept the notion of an Unmoved Mover as possible. However, one cannot show without mere speculation that God in any way, shape or form exists in any possible worlds--let alone all of them, as we cannot even show that there are other possible worlds.
2. Whatever is possible is either contingent or necessary.
This premise is difficult at best to accept, as one cannot verify everything in all possible worlds without being omniscient. We cannot prove that cats don't exist in all possible worlds, and the same with mice.
3. Whatever is contingent can be actualized.
If you are saying that something can be made to exist in other possible worlds, then you may be correct. A daisy can exist just as well in the universe next door as it does here--but it is impossible to truly know.
4. An Unmoved Mover cannot be actualized.
No, it cannot, as it is by definition changeless. However, if something is changeless, surely it cannot change something else? For example, I am the efficient cause of this post. To write it, I would be changed because I have moved. Sense experience, the basis of every argument for God, tells me that if I cause something, I will be changed--relating to physical position, level of energy, etc. Therefore I would suggest that nothing can change something else without changing.
5.Therefore, an Unmoved Mover exists necessarily.
This conclusion contradicts the premise of the argument. No being owes existence entirely to itself, according to this argument, so the being known as "God" should be subject to this logic as well.
With regards to the "Gap Problem", how can one even be entirely sure that the Unmoved Mover is even the Christian God? It may well be one being, but it might not even be a being we have any name for, or that is indeed omnibenevolent--as Aquinas's assumption is that any God equates to the Christian one.