An incrementally-condensed transitive-reduction of a DAG, with efficient reachability queries
Some keywords to help you search for related work: dynamic transitive closure, dynamic reachability problem, dynamic cycle detection.  Because you want to allow both insertions and deletions, this is known as fully dynamic transitive closure.

Alternatively, you might want to consider whether there could be some structure in the particular graphs that arise in your setting that might help you build schemes that are faster than the general case.