You have actually asked a number of questions. I'll try to address a few of them:
1. It's "Normal" to have a charging line running from your TV to your coach. It's best to include at least a circuit breaker in the line within the TV as well as a battery
isolation so you won't have to unplug from your TV when parking connected overnight. In the case of the wiring, the heavier the better, a #8 usually does the trick, but you have to have an equally heavy ground wire as well.
2. If you ever want to run the refrigerator
on 12 VDC you will need a inverter. Depending on how much the refrigerator
and inverter draws in current, you may be able to run the refrigerator
while driving. But that's a big Maybe...
3. Inverter size is dependent on your load, but you may not need a 2000 watt inverter. Check the draw of the refrigerator on it's info plate.
4. As far as batteries, it again depends on the load and how long you want to run off grid. You need to provide some numbers for the math wizards to give that advise. Suffice to say though, if you are going to be mostly off-grid for days at a time, it will take a couple of batteries to keep the refrigerator running all the time...
That's why LP refrigerators were invented for RV's. Because your Amerigo
originally was set up for a 2 or 3 way refrigerator, you might want to rethink going electric. Yes, a dorm refrigerator saves a few $$ up front, but the LP one will run all of the time, no matter what planet you are on. Lots of used ones sell for well under $150
And that's my take.... there will be others