The turn signals are probably quite unrelated, a fuse or other wiring problem.
The radio very well may be powered by the house battery or have a switch to choose which battery it gets power from. This varies quite a bit with various RVs. I would be surprised if the electric seat is powered by the house battery, but I suppose anything is possible.
If you'll generally be leaving your RV plugged in (and thus the house battery charged), one easy way to ensure the chassis battery also stays charged is to install a Trik-L-Start unit. This is a little box that "steals" power from the house battery charger for the chassis battery when needed, and is the sort of thing you install once and then can ignore for the rest of your RV's life.
