Hi Paul
You have succeeded in completely overloading my brain, trying to contemplate God and the infinite all in one question

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So here's my try at getting my brain around it:
If our space-time is infinite, then any possible event (no matter how improbable) has or will occur (yeah, I know you already said this in your question). By extension, in a truly infinite system, there is no impossible event and any/all events have some non-zero probability, and therefore every possible event must occur (either it already has or it will in the future). So if every event must occur in a truly infinite system, and the existence of God is a possible event, the existence of God has either occurred or will occur.
Everyone have a headache yet? If not, my next thought goes like this. If our space-time is infinite and God does not yet exist, he/she/it must exist at some point in the future by the above argument. Now, if God does not yet exist but develops at some point in the future and has the characteristics generally attributed to him/her/it, he will also not be bound by our space-time limitations and will be able to exist not only in the future, but also in present and past.
So in an infinite space-time the existence of God is not only possible it is inevitable. And God will exist throughout the space-time.
Now my head really hurts, and I am putting on my asbestos underwear awaiting responses

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John Theodossy