I would reface the lower cabinets and replace the uppers to avoid a problem, you can get all kinds of great organization things for the base cabinets to make them more efficient.

If you like the present layout then it's more cost efficient to paint or reface. There is always the risk that you might break the countertops. Also challenging to get new ones for perfect fit to existing countertops.

It would depend on the complexity of your cabinet's, if it's pretty standard straight runs, L- shapes and a square/ rectangle island then yes. You need to make sure your new cabinets are the same measures and the one with the sink has a bit of play left to right with the sink to compensate for minor differences in the placement of your sink hole. Any other minor differences in measures from old to new cabinets can be split between overhangs and leaving gaps along the walls which will need to be covered after with backsplash. I would suggest contacting a fabricator in your area to have an installer remove and install the counters. If there's a seam they will need to break the joint of epoxy with a heat pad so the seam doesn't chip and at install there may be minor trimming to the counters Wow after typing and re reading all that it sounds a lot worst than it really is!

I like the layout of the cabinets, can't see changing those from where they are now. I was looking at it again this weekend, and I think that the biggest problem is the backsplash. The botom of the tiles for the backsplash butt up against the countertop, so I'm worried if I try to remove that, that it'll cause the backsplash to come off. I wish there was a way to remove one cabinet at a time from under the counter, and replace them, so I could leave the countertop where it is. Is it possible to do something like that? If not, I might be looking at replacing the top ones, and refacing the bottom ones.

The counters would need to come off in order to get new cabinets in.