tcis - the case you are arguing is well known - its actually called the
"hidden variable theory" - God has access to means we haven't so he can do things we can't.
But a rather famous scientist and mathematican (John Bell) came up with a rather famous (and testable) proof against this being generally so - hence the comments by Hawking.
So simplified to triviality - not even God can make 1 = 2 or find and even number that is odd. As I said its a very clever proof that has stood the test of 2 decades scrutiny.
Asherman My God isn't any more or less a fix all than your your class of infinity. Nor is it a catch all for things I am yet to understand; that is a gray area holding all of the divine, chance, yet to be understood science and other comlicating factors.
Beyond some frameworks of science we have reasoned theoretical models that can't yet be tested (e.g. String or M-theroies). To me few if any such models preclude a directed intelligence pressing the Start button on all of this.
So in answer to your questions from my reasonings:
Q. What was before the beginning?
A. Membranes in M-theory occassionally interacting. We know too little about them yet - barely their edge topology to spectualte more yet).
Q. How did the Universe begin?
A. God nudged two carefully selected membranes with appropriate dimensions into resonance to form a new membrane we now call our Universe. A membrane / membrane interaction could easily cause a Big bang - and as I said - carefully selected and you'd get a universe with our defining dimensions, laws of physics and physical constants tuned as precisely as they are now to support intelligent life eventually evolving.
Q. Why do it?
A. The heart of this is the realisation that the Universe is non determinstic and we have "God Assured" free will rather than a pre determined destiny. So rather than a forgone conclusion the universe has many possible outcomes and God hopes for a certain set, arrived at a certain way. But this is guided and hoped for process not an assured and tightly controlled destination. If it were anyway else an infinite God would be bored by it already and not need to do it. Only a Universe that is non determinstic and built to be beyond his direct control could prove to be interesting because things are at risk. Hopefully more than mice we stand to win in this game if we do things well! The challenge for God as I see it is how little and lightly he can nudge our development along to fulfil his desires with this realm.
Q. God has no rules to follow...
A. God limits himself big time to maintain his interest. The only infinite in our finite Universe is the path we take on our journey forward. If the universe was just a deterministic tool an infinite God wouldn't get much interest from it. A varying, unpredictable, non deterministic reality with intelligent but free beings is infinitely more interesting to an infinite God.
Q. Which model is more elegant?
A. Mine.

Honestly you need to think through the implications of your multiple infinities alot better.
The heart of the trouble I have with these multiple infinities spread through nature is there is little hard evidence of them existing anywhere - so they look man made to fit poor theoretical frameworks to me. I am a (very) finite being observing everything around me looking very finite; large but very, very finite.
I am uncomfortable even with infinite membranes possible within the six classes of M-Theory.
If I was infinite but limited and living in a clearly infinite world the idea of infinities would hold greater credence to me. But the opposite applies for now.
So I dislike infinite time or infinite luck leading to my existence because it makes reality less meaningful to my perception. I dislike infinite multiverses because we have little evidence and few theoretcial frameworks to make this look remotely credible. To pursue this path beyond science fantasy you need a credible theory of quantum gravity spreading across membranes and we are a long way from there (I'd guess 30 years).
So for me Occham's razor says an occassional but highly specific and subtle directed intelligence, not chance, fast-pathed us to where we are today for a specific purpose. To me that is faith not science. But my faith and science interact well and powerfully.
Being a mathematican I dislike seeing infinities bandied outside of maths handled by experts - because infinities are way too prone to be badly missued and missunderstood.
Time being infinite is a hard one. I feel we understand time the most poorly of all our defining properties (even mass). M-theory allows for 10 and 11 dimensional universes, and certainly those with 0, 1, 2, , 4 and even 5 dimensional time charecteristics. An infinite but one dimensional meta-time construct doesn't align with this level of complexity well.
But I am keen to hear your views and reasoning and very open to considering other points of view and modifying my own as well reasoned and supported arguments present themselves. I am not certain God exists, but I feel this is true. My God is a creationist one, and possibly more bound and alot more complex than the simple construct presented in mankind's christian bibles.