If it is the only thing running the fridge will last a month on a tank of LPG.  The battery is a different story, since the fridge control board and LPG detector both use a fair amount of power even if you used nothing else.  If you convert the lights to LEDs and stay conservative with the other 12 volt appliances you can expect 2-3 days on a group 29 deep cycle battery (until you get to 50% SOC).  You can get up to a week with normal use with two group 31 batteries in parallel, or two 6 volt golf cart batteries in series (this is preferred over the 2 batts in series).

Make sure you never use the DC mode for the fridge unless you are on the road, or are stopping for no more than 30 minutes.  It will kill the battery within hours.

The best place to transport?  In the bed of the pickup truck.  Of course, I no longer have a pickup truck, so they go in the back of the SUV as does the generator, which isn't a great place, but its fine and preferable (to me) than in the trailer as I can keep an eye on them.  I usually create negative pressure in the gas tank by pressing in the sides as I tighten the lid; it helps somewhat with smell/fumes.

Once I make camp (even if its only for a night), propane tank, generator and gas tank all go outside and out-of-site behind the trailer, and are padlocked to the trailer frame.

2. What is the typical length of time the refrigerator will run on the factory battery and/or propane tank?

On battery - 6-8 hours; On propane - 30-40 days.

Of course, its very unlikely that anyone would be just running the fridge on propane.  From a practical standpoint, I get about 10 days of propane use in the summer (stove & fridge), and 3-4 days of propane use in the spring/fall (stove & fridge & furnace during the night).