Will the app keep syncing when it's minimized or my Mac is asleep?

When minimized or in the background it keeps syncing on your normal schedule — the rate is the same whether the window is in front or not. While your Mac is asleep nothing syncs; once it wakes, any check that came due during sleep runs shortly afterwards (when sync is on Auto and you're inside your Active hours), so you catch up without pressing anything. If you want an instant refresh after waking, press Sync now.