Why do we select the metric tensor for raising and lowering indices?

To have a good notion of "raising and lowering indices", you need to have a non-degenerate 2-tensor. If it is not non-degenerate, you might send $v^\mu$ to $v_\mu=0$, in which case you can't say that "lowering the index and then raising it back up" is like doing nothing. Being able to do that is of vital importance. A Riemannian manifold comes, by definition, equipped with such a non-degenerate (symmetric) bilinear form, so there is a canonical choice of notion of raising and lowering indices.

If your manifold is guaranteed to carry other non-degenerate 2-tensors, then you may use those as well to raise and lower indices (though the meaning is different from that of raising and lowering with the metric). This is the case, for instance, if your manifold is symplectic (i.e. carries a closed, non-degenerate 2-form $\omega$).

It is a convention$^1$. If a theory has a distinguished invertible rank-2 tensor, why not use it? Examples:

1. The metric $g_{\mu\nu}$ in GR.
2. The $\epsilon_{\alpha\beta}$ metric to raise and lower Weyl spinor indices.
3. The symplectic 2-form $\omega_{IJ}$ in symplectic geometry.

If there are more than one distinguished invertible rank-2 tensor, one would have to make a choice. E.g. bi-metric GR, etc.

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$^1$More formally, the notion of "raising and lowering indices" reflects the musical isomorphism. See also e.g. this Phys.SE post and links therein.

The metric tensor is defined by its ability to raise and lower indices. Take a finite dimensional vectors space, $V$, with $\operatorname{dim}\{V\} = N$. Given another space, $W$, with dimension $M$ you can construct the space of linear maps between those spaces. We call elements from the space of linear maps matrices, and in this example, they form a vector space that has $\operatorname{dim}\{L : V\rightarrow W\} = N\times M$.

If the target space has dimension $1$ (i.e. the map from $V$ to scalars) then the space of maps has dimensions $N$. As we learned from linear algebra, any two finite dimensional spaces with the same dimensionality are isomorphic. We can therefore construct an isomorphic map (i.e. a map that is both 1 to 1 and covers the target space) from the $V$ to $L$. The map from $V$ to $L$ is called the metric.

In other words, when the index is up, the vector is from $V$, when down it's from $L$ (often called the dual-space).

Invertability aside, we also pick the metric to have other properties we desire. In particular, we want the scalars produced by $g(v_1) v_2$ to be invariant under some set of transformations (usually rotations or Lorentz transformations). This is what constrains the metric signature (pattern of signs of eigenvalues).

Also, there is no proof that $g^{\mu \nu} g_{\nu \alpha} = \delta^\mu_{\hphantom{\mu}\alpha}$ because that is the definition of the inverse metric (the map from $L$ to $V$, as opposed to the other way around).