I guess I’m just an old-fashioned dinosauer, but I use good old Elmer’s. Not the politically correct washable, safe, non-toxic stuff, but the original multi-purpose Glue-All.
I save scraps of wrapping paper in different colors and thicknesses to make repairs to boxes. I also make a wooden block that corresponds to the inside dimensions of the box I am repairing. This acts as a mold to keep everything straight and true and is removed before the last piece is repaired. I cut pieces of the appropriate paper to make a “hinge” for loose flaps, or simply to reinforce weak areas. All of the repairs are done to the inside of the box. I make a pencil note to one of the inside repairs noting the date the box was repaired. Using these methods I have reconstructed boxes that were nothing more than a pile of loose pieces.
I remove labels by steaming or soaking, iron them flat, and make repairs or reinforcements on the back. They are then glued back on the box in the exact position they were originally.
Most taped repairs can be removed with solvents such as acetone or lacquer thinner if you are patient. Those same solvents can be used to remove many hand written inked notations, a soft art-gum eraser for the penciled ones. A box of different colored soft chalk, blended with a Q-Tip can hide a lot of blemishes.
Hurrying is your biggest enemy. Plan ahead, otherwise it’s easy to glue yourself into a corner. I make only one or two repairs at a time, letting the glue dry overnight. On a really bad box it may take a week or more to complete. The hot, dry Arizona sunshine helps.