The Converter/Charger maintains charge on your Coach (House, not Chassis) battery. Older C/C's tended to boil batteries in storage. Newer ones don't. You don't say what you have and I'm not a converter expert so that's about all I can offer. But that's the gist of it.
Not sure if you mean while in storage? Unless your using it to top off the batteries for day or so, I'd say no. Why risk any chance of a power surge you don't have to.
I keep our 96 Southwind plugged in 24/7 when it's not in use. Check the water level in your batteries occasionally. No problems being plugged in all the time.