Will the Astroclock on/off times adjust every day?
Yes. Assuming sunset and sunrise time change more than a minute each day, and the sensors communicate with Manage, the time will change each day. Though, there are tons of other settings in groups and profiles that could impediment the change. So, it depends on the profile and group settings.
Do the sensors need to “sync” with Manage to adjust the on/off times?
Yes. Because the sensor needs to sync only once during the 24 hour period, it is not required for the sensors to sync at sunrise and sunset.
What time is used for the Astronomical clock? Is it the sunset, twilight, or last light?
Yes. It should change considering the other profiles, and group settings do not override the change. It is exactly at the sunset and sunrise based on the latitude/longitude available in the facility settings (use the plus or minus 'x' minutes if you want twilight or last light. Refer to the article Understanding Light Profiles.