That’s a lot of questions, but I’ll do my best.
If I’m not mistaken (which I may be), those 12 dimensions you’re referring to are required by String Theory, not abstract mathematics.
The word dimension can refer to a few different things. It can mean the spatial dimensions of length, width, and hight. Mathematicians have come up with abstract concepts of higher spatial dimensions, and have even drawn hypothetical fourth dimensional shapes, the best known of which is the hypercube.
In science fiction and in colloquial usage, other dimensions refer to alternate universes. The many worlds interpretation of quantum mechanics provides some basis for this idea, but it is often misunderstood (I don’t pretend to fully understand it myself). I don’t think it makes provisions for traveling between universes.
Einstein’s Theory of General Relativity also requires four dimensions – the three spatial dimensions and one of time.