The Bloch-Kato conjecture is actually more precise than that. As mentioned by Hunter Brooks, there is indeed an $\ell$-adic Abel-Jacobi map
$$\phi : Ch^j(V)_0 \to H^1(G_K,H^{2j-1}_{\mathrm{et}}(V\times_{K}\overline{K},\mathbf{Q}_{\ell})(j))$$
The map $\phi$ is also called the cycle class map and is defined for any field $K$, say of characteristic $0$.
Now if $K$ is a number field, the conjecture of Bloch-Kato predicts that
(1) $\phi$ takes values in $H^{1}_f(G_K,\cdot)$.
(2) $\phi \otimes \mathbf{Q}_\ell$ is an isomorphism.
In fact (1) is a purely local question : this is really a question about the Abel-Jacobi map associated to a variety over $\mathbf{Q}_p$ (where $p$ can be equal to $\ell$ or not). The conjecture (2) together with the Beilinson-Bloch conjecture implies the statement you mention about the order of vanishing.
You can find a good survey on the $\ell$-adic Abel-Jacobi map and more details about what is known here :
J. Nekovar, p-adic Abel-Jacobi maps and p-adic heights, In: The Arithmetic and Geometry of Algebraic Cycles (Banff, Canada, 1998), 367 - 379, CRM Proc. and Lect. Notes 24, Amer. Math. Soc., Providence, RI, 2000.
See also the "Conjecture $\mathrm{Mot}_\ell$" in the following article :
M. Flach, The Equivariant Tamagawa Number Conjecture : A Survey.
By usual (sometimes not so trivial) homological arguments, one can reduce to the case where $M$ is a finite discrete module over an artinian ring of residual characteristic $p$. In that case, I think you want $S$ to contain places above $p$ as well, even if your $M$ is unramified at $p$, so let me assume this.
The module $M$ induces an étale sheaf $M_{et}$ on $\operatorname{Spec}\mathcal O_{L,S}$ for all finite extension $L/K$. The spectral sequence UPDATE (converging to $H^{i+j}(\operatorname{Spec}\mathcal O_{K,S},M_{et})$)
$$E_{2}^{i,j}=\underset{\longrightarrow}{\operatorname{\lim}}\ H^{i}(\operatorname{Gal}(L/K),H^{j}(\operatorname{Spec}\mathcal O_{L,S},M_{et}))$$
then induces isomorphisms between $E_{2}^{i,0}$ and $H^{i}(\operatorname{Spec}\mathcal O_{L,S},M_{et})$ or in other words $H^{i}(G_{K,S},M)$ is isomorphic to $H^{i}(\operatorname{Spec}\mathcal O_{K,S},M_{et})$. So you can assume that you are working with Galois cohomology throughout $provided$ you use Galois cohomology with restricted ramification.
Because the Tamagawa Number Conjectures are formulated only in the setting above, Bloch and Kato could have used Galois cohomology instead of étale cohomology everywhere without changing anything. To touch upon your last question, I think there are two reasons why they chose étale cohomology.
First, at least at the time they wrote, Galois cohomology was not the most familiar object of the two. In fact, many classical well-known results were given correct complete proofs only very late (in the late 90s in some cases). On the other hand, SGA (and works of Bloch and Kato themselves) existed as references for étale cohomology.
Second, using étale cohmology, one can formulate the TNC over more general bases than $\operatorname{Spec}\mathcal O_{K,S}$ (for instance any scheme of finite type of $\mathbb Z[1/p]$). This kind of generalization had been the key idea of previous works of Kato and Bloch-Kato on higher class field theory so it is not surprising that they decided to at least allow the same kind of generality in their subsequent works.
Best Answer
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 $\mathbb{Q}^{ab}$.