What bounds can be put on counting reachable nodes in a dag?
The transitive closure of a directed graph with $m$ edges and $n$ vertices can be computed 
slightly faster than $O(mn)$ time, but not by much. An $O(n^2 + mn/\log n)$ time algorithm is mentioned in a footnote of Chan's 2005 WADS paper on APSP (journal version in Algorithmica 2008). A slight improvement to $O(n^2 + mn \log(n^2/m)/\log^2 n)$ can be found in the ICALP'08 paper "A New Combinatorial Approach for Sparse Graph Problems " by Blelloch, Vassilevska and Williams.
That being said, I don't know whether counting descendents is easier than actually finding them