The joysticks can be replaced easily, but taking them apart for a good cleaning will likely give you good results if you don't feel like spending the money.
Agreed. Throwing out perfectly fine sticks is just wasteful.
I haven't bought new sticks since I don't know when. You get a ton more mileage out of sticks by just looking after them. Cleaning, greasing, and sometimes I might swap in a new pivot cylinder (90 cents on Focus Attack).
As for snap in versus screw in, unless you're changing buttons often, snap in is the way to go. I can never get the screws tight enough that they don't get loose after a while. Sure, you could glue them like they do in Japan, but then it kind of defeats the purpose.