The Shafarevich conjecture belongs to the broader program of Inverse Galois theory, and in that context it is just another step in that particular approach to understanding $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.

So the answer to the second question is definitely not. For example we could prove the Shafarevich conjecture and still don't know all the finite quotients of the absolute Galois group of the rationals.

Even our understanding of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}^{ab})$ could still be far from complete, since a lot the results are not constructive. For example Iwasawa solved the solvable part of the conjecture in the 50s, but I doubt that we know how to explicitly generate most finite solvable groups over $\mathbb{Q}^{ab}$.

On a side note, you might get a better idea of the impact of solving the conjecture by seeing what we have learned in the one case we have managed to solve, $\mathbb{Q}^{tr}(\sqrt{-1})$, where $tr$ indicates generated by all totally real algebraic numbers ($\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}^{tr}(\sqrt{-1}))$ is a free profinite group of countable rank by results of Pop and others).

The answer to the first question might be no as well, see there other two MO questions in the same spirit (1, 2). Trivially, the Shafarevich conjecture implies the inverse Galois problem over $\mathbb{Q}^{ab}$.