The fridge continues to run off the car battery. I don’t know how it would work as a cooler only. It doesn’t seem as well insulated as a typical igloo cooler. I have the transit bag which adds an additional 1/4” of insulation (and protects the fridge).
Battery life is very subjective based on your battery size/ capacity and wiring. With the second battery in my car, a group 31 Optima yellow top, it can run for 3-5 days without killing the battery when the car isn’t driven. I left the fridge on for a 5 day trip with the car parked at the PHX airport in the spring (probably mid 80’s temps and the car in the shade), the battery was very low but the fridge was still at 34 degrees. On a 115 degree day in Phoenix in my SUV with the stock battery it hit the battery cutoff (set for medium) by 4pm, the contents were still cool enough at about 55 degrees, as soon as I ran the SUV it cooled back down to 34 degrees. In my wife’s Subaru with their ****** 350 CCA battery I was having cutoff issues by mid day at 80 degrees, swapped the battery for a group 34 yellow top and it would run fine all day. On my Dodge diesel truck with 2 group 65 AGM batteries I’ve never had an issue.
The fridge draws about 3 amps. On the SUV and Subaru the stock wiring to the rear 12v outlet is only 18 gauge, insufficient for the load. My car has 8 gauge wire direct from the battery to the fridge plug. The ARB wiring kit comes with 10 gauge wire. The fridge has an adjustable protection circuit, it should be 11.5 volts for the medium setting, but in my SUV and Subaru it shut off the fridge at about 12.2 volts on the car battery. I think the tiny wiring was making the protection circuit see less voltage and activate prematurely. If wired properly off a decent battery it’s fine day to day if the car is being driven. With just a single battery I used to carry an extension cord and my Ctek charger in the car just in case. Sometimes I drive to a location and then drive less than 10 miles a day while on site, not enough to charge the battery in my SUV or Subaru.
The fridge also runs off 110v. If the fridge is warm I’ll usually plug it in to an extension cord for the initial cooldown rather than just let my car battery do it. It cools fine off the car battery, but a little faster off mains power.