For the sake of convenience, I may use "He" as a pronoun representing "God" and "God" in place of "God(s)".
Line-for-line reply:
Depends what you mean by 'real'; tangible, physical?
As a part of mathematics, a science, imaginary numbers, I'm pretty sure, are logical.
Dreams are affected by our experiences and brains.
It's not inherently illogical. Even if it were, what's your point?
Infinity is not logical? How so? Again, is it not also a part of mathematics?
There's no proof there's no physical manifestation of infinity, but even assuming there were proof, I don't think anybody has said there must be one of God. For God to have magically created and to be able to directly, unavoidably change reality, he must have physical influence. Surely, it must exist in some way to do so. Similarly, to be consistent, you should believe in Santa Claus, the Tooth Fairy, the Easter Bunny et alii. Even if there is no physical manifestation of infinity, it would just be a concept, and would therefore exist in the mind, as an abstract concept incapable of direct, unavoidable physical influence.
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This seems to be going in circles. Just because Allah could exist, doesn't mean it does. By your logic, you should believe in all possible gods, including my Teapot God, my Tooth Fairy God and my Santa Claus God.