Wow, that's quite a set of questions. I'll try to handle some of them.
Conservation of probability should certainly hold in any type of physics. It's hard to see what it would even mean for it not to hold.
The main motive for thinking of higher dimensional physics is to find a way to work gravity consistently into the framework of quantum mechanics. So in that program, all the basic quantum properties would survive. That includes the momentum-position uncertainty relations, which aren't tied to any specific number of spatial dimensions. Also, conservation of energy and momentum follow by Noether's theorem from the translational symmetries of time and space. So the time part would still apply.
The proposed other dimensions do not currently behave like the 3 with which we're familiar. In most pictures, they're all currently curled up on a tiny scale, not extended. Some versions, I think, have other extended dimensions but not ones that just mix with our 3 by rotations. So they're different. I'm not quite sure what becomes of conservation of momentum in any little curled up spatial dimensions. You should find somebody more knowledgeable.
The common suspicion is that our current laws of physics reflect a particular way that the extra dimensions curled up. There may be a huge number of different ways (some estimates are about 10500
) allowing many different laws of physics at the level we deal with, all based on some as yet unknown deeper law.
You'd also need somebody more expert to find out whether conservation of charge necessarily works in some deeper physics in which the electroweak force is integrated with the strong force and gravity, regardless of the number of dimensions involved,
(published on 03/15/13)