What is the relationship between oracle Turing machine $M^O$ and Turing machine $M$ (given $O$)?
The definition of $R$ you gave above doesn't make much sense, it looks like you're trying to build an equivalent TM that doesn't use the oracle, but the point of an oracle machine is to provide the TM with capabilities it wouldn't normally have.

For example, if $M$ is a machine with an oracle for the halting problem, then obviously there isn't in general an equivalent machine that can simulate the oracle.

If all the oracles you are using are oracles for decidable languages such as those used in complexity theory, then you can obviously simulate them (just construct a universal Turing Machine and use it to carry out the computation), but that's besides the point because it interferes with time/space complexity.

Nondeterministic oracle machines are exactly the same, the nondeterministic part is the transition function, queries to the oracle are treated as though they were a single computational step that magically happens, it is completely orthogonal to nondeterminism.