Those are great questions. I too had the same ones.
1. The refrig does have its own inverter.
2. I was told it is a 600w inverter by the RV dealer.
3. The inverter runs on the battery when I manually turn it on and if I don't turn it off when I connect up to shore power it will continue to run on the batteries, but then again the batteries will maintain a charge while hooked up to shore power.
4. On dry camping
The efficiency of the refrig will not be any different, the compressor will keep it cooling regardless of battery or shore power, it does not know any difference. 110 volts is basically the same whether from inverter or shore power.
It will continue to make ice as long as the water pump is on or being connected to city/park water
My experience is the refrig will last about 3 to 4 hours on battery but stay cool for 8 to 10 hours, depending on how many times the door being opened. Forrest River does not put very large amp hour batteries in the coach we have so if you install more or larger batteries that could change your decision.
We have a 2013 Georgetown XL 378 and love the residential refrig and will not go back to propane. It cools down to 40 degrees if needed but we keep it at 45 degrees and the freezer easily freezes to 0 degrees. And it does it within 3 hours. Our preference is residential.