Can the laws of physics work with any number of dimensions (whether they be space or time)? That's what Lisa Randall claims, but am seeking clarity. If so, does that mean Quantum Mechanics will still predict the same results in 5 or 6 dimensional universes, and the equations will stay the same?
I'm not able to provide you a definitive answer, but your question reminded me of this minutephysics video which briefly discusses the laws of physics in relation to dimensionality: http://bit.ly/IKjnHR
Interesting video... but what happens to the force of gravity if there happens to be more than 3 dimensions (as is in String Theory)? The implications described in the video say it all.
That is an interesting question. The laws of physics are a bunch of mathematical equations together with a bunch of correspondence rules between the variables in the equations and experimentally measurable quantities. You can certainly write the mathematical equations in a form which is independent of the number of dimensions. But if you then do calculations with those equations in anything other than 3+1 dimensions I think you lose the correspondence rules since we have no experimental access to other dimensions. Once you lose those correspondences I am not sure that what is left should still be called a law of physics.