In reverse order:
If you're hooked up to live AC power (shore, generator, ?) your converter will be applying +12V to the connector. There may be some possibility that something in the truck could be impacted but I can't think of what...
Running a portable generator in your truck's bed while underway could present a problem if the gas tank is full and the road is rough: fuel could spill through the cap vent and be ignited by the engine.
As to having a genset running while in motion, OTR types do it frequently, MH types do it frequently, and I've been known to do so as well.
Of course, I'm not using a portable genset in the truck bed...
I've run our genset in hot weather to supply power to the 2 A/Cs and in cold weather to power the electric heat in the fireplace.