Is it Normal to Pick at Your Body?

I think you might want to look into body dysmorphic disorder :) everything described perfectly sums it up, and it could help to look into how other people with it cope with their habits and symptoms! Even if you don't have it, it's worth looking into as skin picking or things like that is one of the main things with it and it's helped me in the past to look at specific strategies for bdd! Hope this helps at all!

It's not "normal," but I deal with it too. I still haven't found a way to stop it

Most would say its definitely not normal to do that. I myself do pretty much the same as you do. To make it less annoying for me when I feel like doing that stuff I usually just try to listen to music or do something else to keep my mind busy. I just try to find a less harmful activity to do like for example gently brushing my hair.