A RV water heater is safe to leave on. That said it takes a very short time to heat water so you could shut it off during the day and turn it on when you get off work, this would also save a little money.
I use the propane to heat water at the seasonal and shut the heater off of a night so I don't have to listen to it cycle on and off.
It would be great to find someone to walk the dog during the day. I know I would hate to go 8 hours without urination.
Leave the A/C on along with a tv or radio for the dog. This will mask outside noise and the dog will be less likely to bark.
Good luck with your new beginning and remember when one door closes another opens.