A magic eraser on an NES cart is not a great idea, IMO. They are just very fine abrasives, and while you can get the marker off, you will be able to see a difference in the remaining texture. The color is a bit different after as well, but that might just be because of the texture being worn down.
Yes, you can dishwash circuit boards, although I'd think it is not a good idea with a battery backup. I haven't done it myself, but it is essentially the same as on of the later steps in board manufacturing.
edit: these methods have been hashed and re-hashed in the 'Tech and Resto' subforum (very bottom of the page), so it would be a good idea for anyone considering cart restoration to do some searches in that forum. Also, the toothpaste method to clean contacts seems labor intensive. I use a high quality metal polish and it work great for me, and sounds like a lot less work based on your description.