1: As long as you have tags, registration and insurance, there should be no other paperwork required.
2: Always a good idea, as proof of condition before shipping.
3: Once you accept the car and drive it off the delivery yard, they are absolved of claims for damage that you discover later. Always good to do a walk around before signing the delivery receipts.
4: Opinion based, not really allowed on this forum.
5: Depends on the shipping company, some consider personal effects to be extra cargo and charge accordingly. But if it is OK, I wouldn’t put anything of value in there. The car company will likely want keys to car and trunk.