I've been at Changi at 3am before (Had a flight that departed at 3.50am! - Very odd timing)... anyway, the entire terminal is really quiet, there are some food outlets open, but not many... I would suggest actually staying in the transit area and then waiting till a reasonable hour before going through immigration. There are free internet terminals throughout, and if you want to rest, there are lounge seats throughout... though u might notice that staff on night shift also take their naps at these seats.

As for your luggage, I wouldn't really worry too much. 3am is a pretty unearthly hour for a flight to arrive, and Changi would be pretty quiet, hence there really shouldn't be much problems leaving them on the belt or to be collected by the luggage attendants. Just make sure you don't lose your baggage ticket.