[Randy E]

We have a 2017 3791RD with the same fridge.
#1- You can leave it on all the time. I only turn mine off when I want to power off the fridge when it's parked at home. Flip the breaker also to' off' if doing this.
There is a automatic switch that bypasses the inverter as soon as it detects shore power. Then it runs on shore power as long as you stay plugged in. Once you disconnect power, it switches back to inverter power. This is only on the residential fridge.
#2-I have no issues towing long distance and maintaining battery while the fridge is on. Absolutely no problem with the TV powering it as long as it is running. I've pulled for days without issue.
#3 I always look inside my fridge before pulling out on a trip after Shore power is removed.

A word of caution. If you stop for a night at a place where you cannot connect up to shore power, Unplug your TV from the RV. The battery in the TV will drain all night keeping the fridge running. Once you wake up, you might have several dead batteries. Just remember to plug back in before moving. As long as the TV is running the alternator will keep up. Stopping for a few hours to grab a bite to eat, has never caused me issues.