God exists subjectively?

[Tonus]

c. But, if everything are moved, then everything must be moved, which is equal to infinite regress, which is equal to explaining nothing.

There is your exception and your contradiction. In order to break the chain of infinite regress, you must either invalidate the premise or introduce an arbitrary factor that defies the premise for no other reason than to salvage it. Claiming that it exists because it is required ignores the possibility that the premise is flawed. Claiming that it is a specific entity with no explanation or evidence ignores other possibilities that are equally possible (or more possible, when you consider the nature of a God capable of creating our universe).

2. I am not sure if I have gotten your point here properly. Are you saying that we must prove that the quality of God is such and such, before we can prove that there is God?

I am saying that you cannot claim that God is an unmoved mover or an uncaused cause in order to support the premise that would prove that God exists. The equation does not give us a result of "God" unless we define God so that he solves the equation. That's convenient and self-serving. It's like saying that 2 + 2 = 5 because 5 has the intrinsic quality of also being 4, without explaining how this is so. At this point I can provide any number as the answer by assigning it the same quality.