Does syncing happen automatically, or do I have to press a button?

By default it's automatic: the app checks on the interval you set in Settings → Scheduling (default every 5 minutes), and the rate is the same whether the window is in front or in the background. It also syncs once when the app launches. Setting sync to Manual stops the automatic checks, with one exception: if pushing is on Auto, the app still syncs right before each scheduled push (a push always syncs first). You can also sync on demand any time with Sync now, which forces an immediate check. Switching to the window, restoring it from the Dock, or waking your Mac deliberately doesn't trigger a sync — it sticks to your schedule.