Battery recharging is dependent on use. If you run the heater at night you'll have to charge the batteries everyday. The more you use the water pump/ lights/ radio/ ect. the faster the batteries will drain.
The Honda EU2000 is a great generator and will charge your batteries just fine. That generator may or may not run your AC though-- if it doesn't and you want a generator to do the job you can either buy a second Honda and run them parallel or buy a EU3000 or a Yamaha EF24000 (it will start your AC -- but like the EU2000i it will only run the AC if everything else is off). I went the (2) Honda EU2000 route and I usually only carry one.
If you have a battery wizard on your converter it will charge the batteries even faster.
Propane is sort of like the batteries-- while dry camping your fridge and hot water heater will run off the propane. If you run your heater it will also run off the propane tanks. But propane a couple propane tanks should last several weeks.
Your right that cooper pipe does go to a low pressure outdoor BBQ or stove. I don't use mine as I just coonect a 20' hose on my small BBQ -- I don't like the idea of having the BBQ that close to the trailer wall. I like to run it to a little table we bring along.
When I replace our all in one DVD/ surround sound system I'm going to get a Sony DVD player and separate sound system. The one that came with ours is junk. So your not missing much if yours is missing.
Call Jayco and ask them for a manual -- they will send you a generic one. But you'll get the gist of how everything works with it.