And no, most animators do not build there own puppets. I learned all of this stuff out of necessity while trying to make my own films early on. When you can't pay people, you gotta make it do! haha
There are a lot of stop motion animators who are multi-talented tho for sure. My buddy Max Winston for example:
https://www.ratbatstudios.com/
He's a goddamned savant when it comes to this stuff.