That GFCI is obviously not powering those outlets. Either there is another outlet between the GFCI and the four you're trying to fix, or those four get power from somewhere else. My guess would be the former.

Good job on the pics. I would definitely just buy a new door. If you need advice on how to get the right one and how to install it you can DM me. I'll be happy to find a good video or two that will show you the steps.

A photo would help but based on your description it might be fixable. However, interior door slabs are really cheap and I would just buy one, hang it, and paint it.

Based on your description and the photos I think you would be fine removing the framing box and adding another wide stud (you mentioned 2x10). You could probably find a structural engineer who works with architects you could pay for an hour to come look and give you professional advice. IMO it would be worth the cost for the peace of mind. I've done a lot of work like that but I would not modify structural work on a living space without an engineer's input.

For a white bowl we use a Rustoleum product called Whink and it works quite well. Don't use it if the bowl is not white as it will mess up the color.

Yes. Search for apron front sink with backsplash.