First, you most likely do not have an inverter. 120v outlets are standard in RVs for when you're hooked up to shoe power. There may be a 12v "cigarette lighter" outlet in the RV.
Second, it sounds like you had the light on all day. That, in addition to the other pulls will run the battery down. How did you power your water heater - propane? What did you run in the evening/night? What type and size battery do you have?
You may need to check for parasitic draws also. Could be something is pulling power that you are unaware of.
I can't comment on your third question.