Is it always possible to eliminate an edge from a DAG without introducing a cycle? Given a DAG, is it always possible to find an edge such that if that edge is removed and the start and end nodes of the edge merged, the result is another DAG?

Yes!

In order to see this, consider a topological ordering $v_1,\ldots, v_n$ of the DAG and the arc $(v_1,v_i)$, where $v_i$, $2\leq i \leq n$, is the first node for which an arc starting from $v_1$ exists.

edit: note that you might end up with self-loops, i considered these cases to not be relevant.