Memory Requirement for a Computable Problem
Yes, there's plenty of theoretical work on space complexity.  No, it's not necessarily Turing machines; you can talk about space complexity in other models of computation, e.g., the RAM model.  You might find that more practically relevant.  I suggest you study RAM machines and existing work on space complexity.

It's not clear what you mean by "levels of memory", and I don't see anything there that's useful.  Instead, standard space complexity results talks about the space requirement: e.g., $\Theta(n)$ space, $\Theta(n^2)$ space, $\Theta(2^n)$ space.  See also LOGSPACE, PSPACE and similar complexity classes.