Shore power with out battery connected

It would be most helpful if you tell the model and brand of converter. Otherwise all you will get are guesses.

It really depends on what converter you have. Give us its model name and number and we can give you specifics. Most of them are OK with that, but a few aren't.

An '04 RV should have the power supply type converter, so yes it should be fine w/o a battery.

It all depends on what you wish to have it plugged in "for". I would leave the battery on and don't worry about it, you do or will have some items that like to drain the battery "stereo" and such.
Also it isn't good for a battery to be left by it's self. If you are going to disconnect. Remove it and store it in a dry area NON freezing and keep a trickle charger on it.
You will not have lights or most items that run with 12 v if you disconnect. Remember to check the fluids in the battery very month , to save you a headache when you do want to use it and keep it charged.
Good luck