Is this pretty much yours at the end of either the vamp/dawnguard questlines? Are the chests tagged not to respawn meaning stashing gear there is safe? Can you move your family there?

Yep it's pretty much yours if you complete the vampire quest line. Don't know about the dawnguard side but I don't see why not. They are not soppused to respond on the vampire side because you become the head vampire and everything becomes yours, so containers are safe. Like I said don't know for dawnguard side and you can't move family there.

edit: Found out only 38 out of 41 containers are safe here is a link http://www.uesp.net/wiki/Skyrim:Volkihar_Keep but I have never had a problem with storing stuff in the castle before.