Complement of a solid genus-2-handlebody in $S^3$

The answer depends on what you mean by a solid genus-2 handlebody, and the trouble is that this is ambiguous and requires some interpretation.

Had you asked about "the complement of the solid genus-2 handlebody" then I would assume you meant this and the answer would be yes, as in the answer of @QuangHoang.

But since you asked about "the complement of a solid genus-2 handlebody" then I think the answer should be no, because there are many examples of knotted genus-2 handlebodies whose complements are not genus-2 handlebodies. For example, start with a thickened trefoil which is a knotted genus-1 handle body in $S^3$, and then add to it another solid tube to obtain a knotted genus-2 handle body.

Yes, that's correct. To see why, one can think of $S^3$ as $\Bbb R^3$ with an infinity point whose a neighborhood is the complement of $B^3$ in $\Bbb R^3$. It is left to see that $B^3-T\sharp T$ is equivalent to the punctured $T\sharp T$.

Note 1: it may help to work with the solid torus first.

Note 2: It is somewhat obvious in the view of Morse Theory, but I can't think of an obvious Morse function on $S^3$ with $2$ critical points of index $2$ and $2$ of index $1$.