Generally the converter/charger and the batteries and the 12V distribution system are all tied together (at least when the battery disconnect is not disconnected). You can think of it as power coming from the converter to both the 12V system in the coach and to charge the batteries, at least up to its current capacity. If you're consuming more current than it's supplying, the batteries will supply the excess.
What happens with the battery disconnect switch turned off is a bit more variable. On my particular unit, the converter and battery stay connected, but the 12V distribution is disconnected from both of them (so the lights, etc. don't work but the battery charges). I would guess that some have the converter on the other side of the disconnect switch, and probably some more complex variations exist as well. Some (mostly older) converters don't work properly with no battery attached, and others operate but provide very unclean DC power, relying on the battery to act as a filter. Incidentally, some solar controllers also have a similar need to be connected to a battery.
To summarize, the answer to your second question is "maybe."
