Is this a known problem, and is it #P-complete?
To Question 1: The graphs induced by such edge sets are known as pseudoforests. As your proof of the membership to #P implies, an edge set is the image of a selection function if and only if the graph induced by the edge set allows an edge orientation in which each vertex has out-degree at most 1. A graph is a pseudoforest if each connected component is a tree or a tree + one edge. It is known (and easy to see) that a graph is a pseudoforest if and only if it admits an outdegree-at-most-1 orientation.

To Question 2: The problem is #P-hard on general graphs (not sure about bipartite graphs). The family of edge sets of a graph inducing pseudoforests forms the family of independent sets of a bicircular matroid. Counting the number of independent sets of a matroid $M$ is equivalent to computing the Tutte polynomial $t(M;2,1)$. It is shown by Giménez and Noy that computing $t(M;2,1)$ is #P-hard even if $M$ is a bicircular matroid given as a graph.