First order logic, set theory and God

[Belacqua]

Space-time is logically necessary for everything else.

Yes, I think so too. If by "anything" you mean "anything that relies for its existence on space-time." We don't want to beg the question by beginning with the axiom that there is nothing else. 

The first cause arguments work on whether we have to stop with the brute fact "space-time just exists, period, no further cause is required." Or whether it makes sense to argue that space-time, too, is caused. 

The Argument(s) from First Cause are lovely examples of why you can't apply "common sense" to the least common event in the history of history and expect the answers to make sense.

When you say "event," are you referring to the Big Bang? Because that brings us back to a temporal example. Causes considered as essential sequences work as well with or without a Big Bang. Although if by "event" we are referring to the ongoing sustaining of existence -- an event that lasts as long as the universe does, then that's OK with essential sequences. 

Granted, it seems like common sense to say that it's silly to talk about uncaused causes essentially prior to space-time, or actus purus, or what have you -- things which we can't picture and can't have empirical evidence of. Nonetheless, if logic indicates that such a non-common-sensical thingy is indicated, we have to consider it. As you rightly say, common sense may be a limitation in non-intuitive cases.