Your battery is almost certainly dead, as the others have said. It may have a shorted cell accounting for the low voltage. I'd also suggest checking the battery cables carefully; they sometimes tend to get corroded internally and look OK but not make a good connection.
Regarding question 1 -- generally you can connect an external charger to the battery in situ. I would be cautious about doing so with an actual equalization charge (as in around 15V or so) as some of the 12V things in the RV would not appreciate that hight of a voltage and a few appliance circuit boards or other things may be damaged.
Regarding question 2, what charges when the disconnect switch is on and off and so forth: there is some variation there in what the disconnect switch disconnects and what it does not. The unit that charges the house battery (and so also provides 12V power when connected to 120V AC) is called a converter, and it would operate identically from shore power and from generator power. Often the converter is connected to the "other" side of the disconnect switch from the battery, so having the battery disconnected while on shore power means that it will not get a charge. This is not universal, however, depending at least partly on what is more convenient to wire up.
Usually the battery will charge from the alternator if the disconnect is switched off, though I don't know why you'd care to be driving the RV around with the house battery disconnected.
On my '98 Coachmen Santara 315QB, the converter connects to the battery side of the disconnect switch, so it will charge regardless of the disconnect state (but it will not power the house 12V loads when the battery disconnect switch is off). The battery connections are actually fairly easy to suss out, as they're all assembled together next to the battery under the entrance steps. There's a main battery fuse of I think 175A, which connects to the isolation relay (to charge from the alternator and also to provide the emergency start function) and the connection to the generator starter. There's also a connection to a little bus bar with spaces for a few of the little rectangular shortstop circuit breakers. Originally one of these (a little 6A unit, I think) is for the dash radio clock and memory circuit. One (20A manual reset) is to power the entrance step mechanism. One (40A) connects to the house loads and to the converter, with two separate wires attached to the load side. There's room for a fourth breaker, but none installed.
I modified that setup a little on my motorhome, putting in separate 50A breakers for the converter and the house connections, and adding a little fuse panel fed by a third breaker for the loads that are not disconnected: the entrance steps, the radio memory, a battery voltage sense line for a meter I put in, and possibly an LSL Trik-L-Start unit in the future.