Use a furniture style vanity, and no side splashes. I’ve done that in two bathrooms in my place.

Mohawk makes a wax crayon with floor color matches,

That looks like it could be a thermofoil finish. Sanding, priming and painting can be done. There is a company you can send the door to that will fix it.

Zinser and a high quality paint should do it. There are also special primers and paints that you can use to repair melamine and I have had pretty good results with that.

You will never get one piece to match the rest, so be prepared to paint all of the cabinets.

A few comments: Check and see if you need a permit for that size. That is over the size limit for a no permit accessory building in most jurisdictions.

No need for any 16 foot boards, just run your joist in the 12 foot direction.

If I was spending the money to build a shed with the cost of materials this these days, I would have the joists sitting on doubled up two by tens, running the 16 foot length. Those would be sitting on pier blocks dug into the ground. If you angle the ends, you would be able to drag the shed in the future, or jack it up and get it onto a trailer to sell it, or move it.

There are lots of plans for free available on the Internet on how to build a shed that will answer many of the questions you have. Also, since I haven’t seen it mentioned, you should have plastic sheeting below the foundation to keep moisture from rising up and rotting out your framing. I live in the Pacific Northwest, and would use pressure-treated wood for my foundation.

You guys are right on the $!. Put down visqueen/6 mil poly, covered by gravel. Source plastic pallets - cut them in half if you can't go full depth. I got mine for free. My woodshed is full depth pallets on one wall and half depth on the end and opposite walls.