Do all games on GOG now support cloud saves? And is there a way to check in Galaxy that all my games' cloud saves are up to date? I'm about to upgrade to a new computer and I'm wondering if I need to manually move any game saves to the new PC or if I'm free and clear with regards to all my GOG games.

No, not all games are supported. There is a "cloud saves" tag for games that do support it.

No, not all of them. Currently, only the following games support cloud saves:
https://www.gog.com/games?feature=cloud_saves&sort=bestselling&page=1 For games that support cloud saves, there should be an indication next to the game's version that the saves are synced (see attached image). Cloud saves are a very recent feature on GOG, so for the most part, you'll still need to manually move your saves around (unless you are lucky enough to only have games that support cloud saves).