Does an algorithm exist to efficiently maintain connectedness information for a DAG in presence of inserts/deletes?
The problem you have described is fully dynamic DAG reachability (also referred to as fully dynamic transitive closure on DAGs).
It's called fully dynamic since people also study the versions where only deletions are possible (then it's called decremental reachability), and where only insertions are possible (called incremental reachability).

There are a few tradeoffs between the update time and the query time. Let $m$ be the number of edges and $n$ the number of vertices.
For DAGs, Demetrescu and Italiano (FOCS'00) gave a randomized data structure which supports updates (edge inserts or deletes) in O($n^{1.58}$) time and reachability queries in O($n^{0.58}$) time (node inserts/deletes are also supported, in O(1) time); this result was extended by Sankowski (FOCS'04) to work for general directed graphs. Also for DAGs, Roditty (SODA'03) showed that you can maintain the transitive closure matrix in total time O($mn+I·n^2+D$), where $I$ is the number of insertions, $D$ the number of deletions and of course the query time is O($1$).

For general directed graphs, the following (update, query) times are known:
(O($n^2$), O(1)) (Demetrescu and Italiano FOCS'00 (amortized), Sankowski FOCS'04 (worst case)), (O($m\sqrt{n}$), $O(\sqrt{n}$)) (Roditty, Zwick FOCS'02), (O($m+n\log n$), O($n$)) (Roditty, Zwick STOC'04), (O($n^{1.58}$),O($n^{0.58}$)) and (O($n^{1.495}$),O($n^{1.495}$)) by Sankowski (FOCS'04). 

Obtaining a polylogarithmic query time, without increasing the update time too much is a major open problem, even for DAGs.