Eternal means forever, and again implies time. In your hypothetical realm that was created to accommodate the deity which has no time to speak of, eternity is equally meaningless - no difference between a nanosecond and a trillion years.
As 'time' itself wouldn't apply, change cannot happen without time, and therefore, nothing that the deity did would / should take any time, implying that everything that it did, it did in an instant, and not over a course of days: "On the first day... rested (a time-based non-action) on the seventh day," etc.
It can and it can't. It depends on the use. In the theological definition it means not bound by time, not in time, not finite, etc., more concisely: "outside time" or transcending time..
I'll address your religious points in a separate reply.
It's an interesting "oh by the way" that St. Augustine, whose Greek was less then perfect, while translating the Greek Old Testament book of Sirach (18:1), wrote "qui vivit in aeternum creavit omnia simul" which means (he) who lives in eternity created everything at once whereas the Greek reads together.
This, of course, clashes with the story of Gensis, so it is interesting to read his rationalizations (in his Creation Days), based on his own mistranslation, in an attempt to "harmonize" the discrepancy between Sirach and Genesis, implying the six days of Creation was really one day repeated six times because "six is a perfect number"!
Of course, the Catholic/Orthodox Church considers Sirach as scripture and a contradiction between them is not an option, so he had to invent something that would make sense in this regard without dismissing either source.
This is, in some ways, exactly what Augustine did with his infinite regress arguments "proving" that God exists as a logical "necessity." even if the argument itself is logically self-refuting.
The only logical thing about his argument is that if anything that exists must have been caused to exist, then that which caused existence things to exist cannot exist (by definition). Which leads to the conclusion that nothing can possibly exist, which is refuted by observing the extant universe.
The paradox is then "solved" by assuming that the first cause is uncaused, and "exists" just because, for no cause or reason whatsoever, as a logical necessity.
Bit if we can assume something like that, we can just as easily assume that energy and time existed all by themselves for no reason whatsoever and are engaged in a repetitive cycle of creation and extinction for no reason whatsoever, without a beginning or an end, like points on a circle, none being the first nor the last, nothing being the beginning nor the end, one always being the cause of the one ahead of it, a perpetuum mobile, a perfect recycling machine or a self-sustained organism.