My experience with fridges...
-Cigarette lighter style will fail and/or unit will come unplugged often. And the issue of dropping / stuffing things in your car with the plug sticking out the side panel
-7-8 amps is the peak draw, usually when compressor turns on, more likely the constant draw while running is 2-4 amps.
-Hardwiring is the best, and if you add a fuse block to the rear, it's expandable for charging points, lights, other accessories, etc
-I wouldn't trust the OEM tiny wires that power the OEM cigarette plug in the rear, 7amp on those little wires seems like a lot.
-Single Group 31 AGM battery is enough to keep my cheap edgestar running for at least 24 hours even in hot weather.
-11.5-12V cutoff on the fridge would be really nice. Mine is standard at like 10.5v which next to useless if I forget and leave my fridge plugged in for days and don't start the car.