Is there a way to lift 6 functors on constructable sheaves to the dg world?
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If I understand correctly your are looking for a dg enhancement of the six operation formalism. There seem to be a paper that does something very close to it: Yifeng Liu and Weizhe Zheng, Enhanced six operations and base change theorem for sheaves on Artin stacks (available at http://math.columbia.edu/~liuyf/sixi.pdf). They use the language of $(\infty,1)$categories, but I think one can adapt it to dgcategories (assuming that one is working over a field of characteristic zero). EDIT Nov. 27, 2012: the above preprint has been posted on the arXiv: http://arxiv.org/abs/1211.5948 


Yes, not just presumably but definitively. 

