In addition to your comprehensive list, I would also inspect : the jack (s) for proper opperation; tires for overall condition and date codes, including the spare; the condition of a jack and lug wrench if they are included; functionally check the vents in both positions and sealing when closed; look at the wiring going to the plug for cracked insulation or broken wires.

Check each window for proper opperation, including the latches and any inside indication of leakage

Plug the trailer into your truck and inspect all the lights, exterior and interior, and brakes for proper operation.

You can tell a lot about the condition of the axles by observing the condition of the tires. If they are worn on just one side, which may include cupping, the alignment is off. This means there are issues with the axle placements.

If the tires are worn in the just center or on both of the outside transitions, an improper air pressure was used.

Take a step ladder with you to help with your inspection. A large piece of cardboard will help with your under carriage examination. A pair of coveralls and gloves may be handy as well.

BOL