Full disclosure: I'm not a current HX Stomp owner, just a prospective one.
But to answer your 2nd question: as I understand it, you can just use a cable splitter to connect the HX Stomp's outputs to your power amp as well as to the FOH, as both accept line level inputs. You can then use presets with names beginning "DIR" (for direct) as these use a full signal chain including amp and cab modelling.