Hence, being extra dimensional compared to us, God would be beyond our understanding.
It is not completely impossible to understand extra dimensions, it is just... weird. But given that humans are so intensely focused upon current dimensions. Which, to be fair, are far more than just three, x, y, and z have no ability to describe how hot something is... Understanding the familiar dimensions is vital to understanding humans, and in order to perceive a dimension you need to interact with it. In order to understand humans you need to interact with them, at which point they can perceive you. If god does not understand humans then god's wisdom, no matter how perfect, will not work to satisfy humans. Any such effect would be purely co-incidental, which demonstrates that god if god is benevolent, and does not exist in our dimensions, then god either lacks the power to exist in our dimensions, or is a fool who chooses to meddle in things that it does not understand. If god is not benevolent, then I would rather a just fight without hope than a life without worth...
The fact that it can be made conceivable, doesn't mean it exist. This is why math doesn't work alone. Its a great tool, but alone proves nothing. As it stands, extra spatial dimensions exist on as mathematical proofs that hold up well to other mathematicians.
nobody has wrapped a tape measure around the earth, does that mean that you would doubt a mathematically derived value? Nobody has measured the distance between the earth and the sun, does that mean that we don't know it? Mathematics DOES work on it's own, that is the whole point. A mathematical argument will always be precise and accurate, the problem arises in translating the world into maths, maths into the world is pretty reliable...
Mathematically proving that something 'can' exist is about as good as you will get for proving that something is possible. It does not, however, prove that something 'does' exist. But what it can do is model them, if you have a working model then you can observe that model and look for any effects it has that are not consistent with a world in which it doesn't exist. If you find these then you can provide compelling evidence for something. Of course providing complete certainty in anything is pretty much impossible...
Acting upon things you do not understand is foolish and should be avoided in all but the most desperate of circumstances. If you do not understand a hole in the ground, if you do not know anything about it, if it simply appeared in a manner that you cannot explain, then you should not stick your head into it, no matter how friendly the book you found lying next to it says that it is, unless you cannot tolerate the alternative, there is every chance that the result is worse than anything possible in you world. If you do not understand god, then it is foolish to allow it to dictate your actions, no matter what you are told by an army of people who also do not understand it, or what it supposedly has people write about it. You are completely ignorant of god's intentions, motives, actions, or the fact that heaven is god's gaping maw that devours its favourite parts and tosses what is left of you into a rotting heap, probably, assuming that god exists, and to be fair it is only because god doesn't realise that you might have an opinion on the subject, and it isn't really like that, it is just a visual aid to help you with something that is beyond your comprehension...