This is a two-part question:
What causes those flaps of skin below and to the side of my thumbnail? What is the reason for them?
At the base of each fingernail, there is a membrane of skin (someone here must know the name). During my childhood (in the 70s), I pushed some of them down to the base of my nail (just because i could) but they have never grown back to their original shape. Can they ever grow back?
The membrane of skin is called the cuticle. The purpose of the cuticle is to seal off the nail bed from bacteria.
I think that opinions may be divided on whether you should push the cuticules back or let them be. If you’re going to push them back, do it after a bath, when the skin is softer. You can get cuticle cream, oil or gel, which helps soften them further so that they can be pushed back with a orange stick. You shouldn’t cut bits off them with clippers unless you’re sure you’re not going to damage any living tissue.