Relating memory complexity and decidablity Given a language $L_u$, about which we know that there exists a non-deterministic turing machine which accepts it (as in, implying $L_u \in RE$) with memory complexity of $c^{p(n)}$, where $c$ is a constant and $p(n)$ a polynomial, can we decide whether $L_u \in R$ or not?

Memory complexity represents the largest difference of indices of used slots of the tape. So I believe $L_u \in EXPSPACE$, although here it is defined using a constant, whereas most articles use a 2 to the power of some polynomial. This is why I'm not sure about that fact. 

Basically, what I'm asking is:

Is $L_u \in EXPSPACE$?

If it is, can this fact be used to derive whether $L_u \in R$, and how?

If not, how can the fact that $L_u \in RE$ and that the turing machine has a memory complexity of $c^{p(n)}$ be used to determine whether $L_u \in R$, and how?
Any language accepted by a non-deterministic Turing machine is decidable, since non-deterministic Turing machines can be simulated by deterministic Turing machines. See for example the Wikipedia article.

There is some subtlety you might be worried about: the non-deterministic Turing machine might always halt (on a given input), but its running time could be unbounded, and then the trivial simulation which just tries all possible guesses wouldn't halt. Fortunately, König's infinity lemma shows that if the running time is unbounded, then there is some infinite computation, so the running time must be bounded.