Can any problem that runs in exponential time on a deterministic RAM be run in polynomial time on a non-deterministic RAM?
We don't know, but probably not.

You shouldn't equate nondeterministic polynomial computation with deterministic exponential computation. A problem being NP-complete does mean that we probably don't know anything better than an exponential deterministic algorithm for it; however, there are a great many exponential deterministic algorithms for which we don't know any nondeterministic polynomial algorithms (e.g., anything that's EXP-complete).

There's no particular reason to assume that EXP$\,=\,$NP. In particular, exponential-time algorithms are allowed to use an exponential amount of storage, but any computation path of a polynomial-time computation can only use a polynomial amount.