There's an airport bus which runs through the night.
Tgere is also a VIP car you can hire to take toto/from the airport ($250 a trip)
AS for hospitals - it depends on the situation.
(a) catch the ferry into town and go to Queen Mary in PokFuLam, or to Central Hospital, OR
(b) call an ambulance, which will take you to Princess Margaret Hospital in Kwai Fong (about 25 minutes bu car)
We have a great doctors service in DB - one (or maybe both?) of the clinics offer a after-hours call service, and the doctors are usually very happy to answer questions about possible emergency situations.
Hope this helps - PM me if you want some more details.