Directed NP-hard problems on DAGs
This only aims at partially answering the first question of the post:


  What are some directed problems that remain NP-hard on DAGs ?


In [1], a few natural problems on directed graphs are given that remain NP-hard on DAGs. The motivation of the paper is to find a "good" treewidth-like measure for digraphs. They argue that the drawback of many measures for digraphs is that they are constant for DAGs, but many directed counterparts of natural problems remain NP-hard on DAGs. For more on this topic, see also [2] and the references of these papers.

[1] Robert Ganian, Petr Hlinený, Joachim Kneis, Alexander Langer, Jan Obdrzálek, Peter Rossmanith: On Digraph Width Measures in Parameterized Algorithmics. IWPEC 2009:185-197. Full version

[2] Robert Ganian, Petr Hlinený, Joachim Kneis, Daniel Meister, Jan Obdrzálek, Peter Rossmanith, Somnath Sikdar: Are there any good digraph width measures?. IPEC 2010, to appear. arXiv