I’ve been really happy with this one so far: https://www.amazon.com/dp/B08ZNRG5NQ/ref=cm_sw_r_cp_api_glt_fabc_BKNYD7810F91XT4QNX78?_encoding=UTF8&psc=1

Yes. I use this paper: https://www.onlinelabels.com/products/ol177wj

Zitco is also super popular, and probably more popular: https://www.amazon.com/Premium-Printable-Vinyl-Sticker-Paper/dp/B08ZNRG5NQ

And then I coat it with a laminate to make it more scratch and water resistant.

3M make a cold laminate which is very popular. Oracle does as well. You'll want to add some sort of laminate to any sticker paper to keep it more durable against scratches and make it safer to hand wash on a cup.