No problems running gen while towing. We do it to power the ac when it's really hot and want the camper cool when we arrive at the campground.
As far as the gen powering the house, I wired a second twist lock plug on my rv tied to the genset. I then use my 50 amp trailer cord to plug into the 50 amp outlet I have near my RV parking space. Normally I use this outlet to power the trailer, but I have a manual transfer switch I can use to disconnect from the street and use the outlet to reverse feed the house. Not as seamless as a whole house generator and auto transfer switch, but works fine and lets me use what I already had.