Finding vertices for which there either exists a path to all other vertices or other vertices have a path to them
We can assume that the DAG is connected, since otherwise the solution is trivial. Consider some topological ordering of the vertices. A vertex $v$ satisfies your condition if (1) $v$ can be reached from all vertices preceding $v$ in the ordering and (2) all vertices following $v$ in the ordering are reachable from $v$. We can check conditions (1) and (2) separately.

To check condition (2), traverse the topological ordering in order, and for each vertex encountered, remove the vertex. Since this is a topological ordering, when a vertex is removed, it is a source, that is, it has no incoming edges. Condition (2) is satisfied for the vertex iff it is the unique source at that point. Condition (1) can be checked similarly by traversing the topological ordering in reverse.