re: Interior of cabinets sweating.

Several things to consider.

1) Could be poor sealing of penetrations in the wall/cabinet allowing moisture from the wall cavity in (many trades rough in with a hatchet it seems). You can do whatever you want demo wise inside the cabinet, seal and insulate, and then trim it out with a panel inside the cabinet if you want to seal this up.

2) Any chance of a drain or plumbing leak adding excessive moisture in this closed space?

3) Consider undercutting the cabinet doors or using shims/stops to make the doors stay open a 1/16" or so to allow more conditioned air to temper the space in the cabinets.

Hope this helps.