Our RV came with a Xantrex Prowatt SW2000 Inverter factory installed. It's only connected to our "residential" style fridge. The setup also included a Xantrex transfer switch, 4 house batteries, and one chassis battery (to start the rig).
The only time I run ours is when we are traveling. I typically run the generator too, but until I recently replaced the carburetor, it wasn't very reliable.
I shut the engine and generator off when refueling. By leaving the inverter on, the transfer switch keeps power to the fridge from the inverter.
Once we are connected to shore power, I typically shut it off. But I have experience power outages at campgrounds, so I might start leaving it on until we get home.
Leaving your inverter on won't hurt it, but it can waste battery power if you don't need it. As for the wear and tear on the inverter, I have this same model connected to my home solar setup. It's been running 24/7/365 since 2012.