In addition to what others have said, Spidey grabs the web in his hand after he shoots it. He’s not actually swinging by his wrists, he’s holding it like we would hold a rope.

Also, calling himself SpiderTeen would weaken his secret identity, as that’s a much smaller percentage of the population. Especially when he gets older and starts calling himself Spiderman, his enemies could easily do the math and start searching for people around that age.