The flaps of skin are probably your vaginal lips. They look more flappy (like fat lettuce hanging out of a hamburger bun) when you mature than they did when you were a little girl. It didn't come from the tampon use, it came from puberty. So even though they are not familiar looking from before, they are normal.
The two strings were attached to the tampon, and are there for pulling it out. (And a good thing, too, it is a pain to lose the tampon strings inside the vagina and have to fish around for them.) If a tampon hurts on pulling it out, it is because it is not very wet with menstrual fluid. On a light day of your period flow, a tampon won't get filled. The cotton sticks to the walls of the vagina, and pulling it out can be a bit of a twinge. On the other side of the coin, sometimes a tampon can get saturated and you don't know, and it can leak through. Many women's most embarrassing life moment had to do with an overflowing tampon and white pants or shorts, and being out in public and not knowing they had a red crotch.
Tamps are great with the caveat that you don't ever know how full they are for sure, so you sometimes wind up yanking out a dry one and sometimes overflow. With time, you will be able to assess roughly when they should come out. A lot of people find that their first three days are the heaviest of their flow, so they change the tampon every time they pee just to be sure it won't overflow. Then on lighter days, their either change to lighter tampons (in other words, ones that are not much bigger around than a pencil, marketed as "lighter flow" or "junior") or they don't change the tampon as often.
While you're getting adjusted to how often to change them, you could wear a light pad as well.
If you are curious about what a normal vagina's lips look like, there are a series of illustrations on the internet that I think you can find by googling Betty Dodson, the name of the artist who drew them. She was just an artist many years ago, and did a graphic series (for political reasons, I think) with illustrations of women masturbating, and the gallery was overwhelmed with letters from women asking how to masturbate. So she kind of lapsed into a second career about women's issues related to their genitals. Her website has a lot of illustrations of vaginas, and they all look kind of wild. Like butterflies and stuff. You needn't fear if your labial lips hang down, guys love labia of all types.
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