Does Storage Studio quit an app before offloading it?

Short answer: Yes. The app is quit before it's archived, so save any open work first. A progress card shows while it's copied to your fast drive.

Yes — the app has to be closed to be archived, and you'll see "Quitting …" in the status. Save any open work in the app before you click Offload, just as you would before quitting it yourself.

Then it copies the app and its data to your fast drive and shows a progress card until it's done.