According to Thomists, it is Existence itself and it has no spatial dimensions at all.
Its Essence is Existence and that's all there is to it, except of course that it also rewards and punishes people, creates things ex nihilo etc.

That in itself seems incoherent to me, but I am focussing on something else here, for which I accpet, for the sake of the argument, that it is coherent.
The question in that case still is: how can such simple and immutable being act differently across possible worlds?

That may be true, but I am willing to grant, for the sake of the argument, that it makes sense.

It is not a matter of wanting to understand God's actions, it is a matter of analyzing claims about his actions in light of logic.

Count Timothy van Icarus

"Thomism suggests that God, in His simplicity, is not composed of separate parts or aspects that might conflict."

But that's the problem. God's intention to actualize A does conflict with God's intention to actualize B.
So, ther can be no intention to actualiz A or B in God's mind. How can God have control over whther A occurs in that case? If God's will is is unified and consistent, then it cannot lead to A in one possible world and B in another, at least not if God is supposed to be in control.

Yes, simple and immutable exclude all properties, but Thomists tend to translate that as "God is identical to all His properties" they say that God's omniscience is His omnipotence is His love etc. and that they are all identical to God. That means they are no real properties, of course.
God is God, and that's it,
But, IMO, this entails that God also is His intention to create X, hence X is necessary.

I know about the via negativa, but I don't tjink it applies to necessary things."God is identical to all His 'properties' " is very clear. Those 'properties' or 'predicates' themselves can be attibuted analogically to God, that is, God's knowledge is identical to God's Goodness, but that doesn't mean that creaturely knowlegde is identical to creaturely goodness.
But i see no way out of the conclusion that if God is identical to all of his properties, then God's 'property' of creating A is identical to God's 'property' of creating B.

I don't know whether I am presenting Spinoza's critique, but what is wrong with my critique?
Additional knowledge seems to be impossible, because then there is no simplicity anymore.

I don't think God can be in everything is we are free, unless we are, in a way, God.

Gregory

I know that lots of Thomists claim that Divine Simplicity does not preclude God's free choices., but my question is how this is possible given that God is identical to all of His 'properties', which would include that he is identical to both His choice to do X and to His choice to do Y and even to His choice to do nothing at all.

I would think God's love for himself is necessary, but then I think everything God does is necessary.

As far as I know, God was not hidden from Adam and Eve and God was not hidden from Satan or any of the fallen angels.