Concerning your first question, every edge-critical graph without isolated vertices must be vertex-critical, but not vice versa. For instance, the complement of a $7$-cycle is vertex-critical but not edge-critical.

Concerning your second question, every vertex-critical graph must be contraction-critical as well. Suppose we are contracting an edge $uv$ of a $k$-vertex-critical graph $G$. Since $G$ is $k$-vertex-critical, there is a $k$-colouring $c$ of $G$ in which $u$ is the only vertex coloured $k$. This is also a proper colouring of the graph with the edge $uv$ contracted.