Residential Refrigerator and Inverter Questions

[RMcNeal]

We have the 3791RD with the Samsung residential fridge. Our dealer DID NOT install a second battery and unfortunately, I didn't know any better when we purchased our unit to question it.
We normally do not travel far each day, so until recently, I would just secure the fridge and not worry about running on the inverter. Everything stayed cold for our 2-3 hour travel day without power. I recently started going a little further now and with summer time getting here in the south east, that really isn't going to work anymore. I have since turned on the inverter and run it while towing so the fridge will stay cold.
Question 1: Is it OK to leave the inverter turned on all the time or do I have to remember to turn it on when towing and off when on shore power?
Question 2: Will my TV (2014 F350) sustain the battery enough while towing with the fridge/inverter running?
Question 3: What is the best way, short of climbing inside the rig and opening the fridge, to make sure it is still running?
Question 4: A bit off topic, but, would replacing our single battery with 2 decent 12v deep cycle ~100 AH batteries be enough? We NEVER boondock. I have the solar prep only, no panels, etc. I do not carry a generator with me at this point, but I am considering a small inverter/generator like the Honda or Harbor Freight one that is pure sine wave.
Any advice appreciated.