Can only DH into a live leg via a Company scheduled pairing. They are not the norm, but recovery pairings built only as needed (and probably almost all done by Reserve pilots)
Every other DH pairing has a legal rest built into it. That being said, there's been a few times when Wx or Mx issues have resulted in my travel being delayed and not having a legal-8 hour domestic rest. FAA doesn't care, and the big concern with the company is are you going to be there for the trip? Are you going to call in fatigued.
My answer to the fatigue question is that I don't expect to, but won't make any promises until just prior to actually reporting for duty