No, subtitles aren't synchronized automatically when downloaded.
I think it's best to try to download the correct subtitle with the release of your video. That way, you should only have to use the synchronize option when there isn't a synced one available.
FlixTools tries to download the right subtitles but if it downloaded the wrong one (correct but out of sync) I would pick another one from another sync group. (The different sync groups have the different color dots next to each available subtitles)
When they're all wrong and out of sync, only then I go through the hassle of synchronizing.
There is no batch mode and Save All right now. However (if your machine can handle it) you can analyze multiple videos at the same time. A shortcut is arrow down through the list and then Option-A when the subtitle is fully loaded.
Hope that helps.