Bathroom sink

Barkeepers Friend. Let it soak for about 30 mins, then scrub with a Magic Eraser.

It looks like a prefabbed sink...not knowing the composite will be difficult to answer

Try WD-40 Spray directly on area let soak for a few minutes and wipe

There is a product, can't remember the name, got it at Lowe's that is specifically for sprucing up vanity tops and cabinet tops. Mine was scratched up pretty bad and I didn't want to use anything abrasive that might make it worse. This product kind of filled in the scratches and made it shine again. I'm sure you can find these type products in a Google search. Best of luck.

Take a photo to the paint department of the home improvement store and ask for their advice. They will sell you the best paint product and application tools and give the best advice on application.

https://www.thespruce.com/the-five-best-diy-countertop-refinishing-kits-3955792

Try Gel Gloss, it's for polishing fiberglass surfaces and does a good job on cultured marble. Comes in spray or can, works kinda like car wax. For the stains, ask your home improvement store.

Rustoleum tub and tile paint kit. Prep is the key.