It looks like Galaxy syncs the saves to the cloud. But on store page there the cloud saves are missing. So are the cloud saves working?

Looks like it. Go into the extra and clouds saves are at the bottom. You can download them.

On a sidenote, game eats some savegames after saving them and when restarting. Happend on 2 machines now. First though I had missed the save button before quitting, but now a second one is gone with 40+ years. Older one (24 years) still there.

Same seems to have happened to me. I manually saved my game twice during a 5-hour session (with different filenames), but when I came back and wanted to load my game, the Load list was empty.

EDIT: It seems the files themselves are still in the Per Aspera\SaveGames folder, but the game doesn't present them when I try to load them from the main menu.