I am sure this is well known, but I am not an expert...so I appreciate any help

Let $C \subset \mathbb{P}^3$ be the complete intersection of two hypersurfaces of degree $d_1$ and $d_2$. Let $U_{d_1,d_2}$ be the open loci, in the appropriate Hilbert scheme, that parametrize those curves. Hartshorne (Serre?) established a correspondence between this Hilbert scheme $U_{d_1,d_2}$ and the moduli space $M$ of some vector bundles of rank two in $\mathbb{P}^3$.

$\textbf{Question 1}$: Is this true? Where can I read about it?

This implies a compactification of $U_{d_1,d_2}$ will give us a compactification of the moduli space $M$ of those vector bundles.

$\textbf{Question 2}$: Is this a standard procedure for constructing meaningful compactifications of $M$ ?

$\textbf{Question 3}$: What is the situation for elliptic quartic curves ?

Thanks!