Some questions about the Computability of Turing Machines
The answer is "we don't know without more information" for all 4.

Suppose $f$ is the identity function $f(x) = x$. Then $f \circ g = g \circ f = g$, which is non-computable.

On the other hand, suppose $g$ is a total non-computable function (which is a special case of $g$ being a partial non-computable function), and suppose $f$ is the constant zero function $f(x) = 0$. In this case:


$f \circ g$ is the constant zero function, which is obviously computable.
$g \circ f$ is also a constant function. We don't necessarily know what the constant is, but that doesn't matter; every constant function is computable. See How can it be decidable whether $\pi$ has some sequence of digits?


Deterministic vs. non-deterministic Turing machine doesn't affect the answer, because a deterministic Turing machine can simulate the execution of a non-deterministic Turing machine (potentially with an exponential slowdown) by dovetailing.