You have many options, but I would not consider manila folders one of them. If you feel compeled to use a box, get an archival box and interleve each print with archival paper.
For mounting it really depends on the size of your print and look you're trying to achieve. Dry mounting is one option, but I would never use it with vellum type papers.
Hinging is another option when used with an overmat as are corners, but it depends on the size of your print and the paper.
I generally prefer an overmat, but I know many people that don't. Lots of folks don't like dry mounting because it's rarely reversible. Dry mounting a print leaving a 1-2" border on the print that isn't covered by the dry mount tissue creates an interesting effect.
You have many options you just need to decide what suits your style.