there is discoloration and scratches in the sink
Yes, you can redo it https://homeguides.sfgate.com/restore-porcelain-kitchen-sink-93752.html
There are kits to resurface porcelain and companies that do so as well.
Yes, I had my porcelain tub painted and it turned out great and is holding up very well. I would contact a professional but if you have nothing to loose then try a product like this one first. https://amzn.to/2YwozXT