I'm running DAVdroid 0.8.4.1 from F-droid on Android 5.1.1 on a Sony Xperia Z3 (stock ROM). DAVdroid will manually sync with Fastmail's caldav server but it will not automatically sync.
The account settings all say automatic sync is enabled but it doesn't matter if I set it to 5 minutes or 4 hours, it always shows the last sync as the last manual sync and the calendar is never updated. It also does a single one-time sync if the phone is rebooted and then never syncs again (unless you initiate a manual sync). This really limits the usefulness of DAVdroid.