Complexity of two perfect matchings with minimum shared edges?
Consider a 3-regular graph $G = (V,E)$.

If there is an edge colouring of $G$ with 3 colours, then you have a partition of $E$ in 3 disjoint perfect matchings. In particular, you can find 2 disjoint perfect matchings.

Conversely, if you can find two disjoint perfect matchings $M_1$ and $M_2$, you can also find a third disjoint perfect matching $M_3 = E \setminus (M_1 \cup M_2)$. The partition $\{M_1, M_2, M_3\}$ of $E$ is an edge colouring of $G$ with 3 colours.

Deciding if a 3-regular graph has an edge colouring with 3 colours is NP-hard (Holyer 1981). Hence it is also NP-hard to decide if there are two disjoint perfect matchings (i.e., to decide if the minimum number of shared edges is 0).