Also, in the "for what it's worth" department, starting a gas engine and bringing up to temp without actually driving it can actually cause more issues than it solves. You'll end up with condensation in parts of the engine and exhaust that you don't want. The engine is better left off until you are going to drive it. That is not true however for the generator. The genset should be exercised monthly under a load for an hour or more. Make sure you disconnect shore power first before doing this. Although most transfer switches can be trusted to convert power from shore to genset without issue, for the 30 seconds it takes to unplug why risk frying your electrical system.
As far as the battery, once you find out why it's not charging or disconnecting, you should get your battery bench tested to see if it's shot. Batteries don't like to be discharged less than 30% and won't really recover fully again.
Once I figured out how the system worked on our second MH, I hooked the chassis battery to a trickle charger once a week overnight to keep it topped off if my trips were less than a few weeks apart. Phantom draws without recharging will kill it.