How to handle an undefined case with µ-recursive functions?
$\mu$-recursive functions are partial functions because of the minimization operator. Without minimization, you get primitive recursive functions, and they're total. So an undefined value can only arise because the minimization operator is involved somehow. For example, you can define
$$\mathrm{undef} := \mu(1)$$
where $1$ is the constant function (with arity $2$) whose value is $1$.

In general, to prove that a function is µ-recursive, you proceed as you would to prove that an object is a member of a class. You demonstrate that the object can be built according to the rules of that class, possibly using objects that other people have already proven to be members.