Delighted to hear of someone actually visiting Northern Germany!
I have flown out of Hannover many times and cannot remember any special difficulties.
We hate these early departures
Absolutely, I totally agree. Especially since they usually tell you to be there three hours before departure for international flights. What's the point in spending money on a hotel if we have to get up at 2 a.m. to catch the S-Bahn at 3.30 a.m.? Irony: last time we flew out of Hamburg at 8 a.m., and it turned out the check-in counter wasn't even open before 6 a.m.! In case you have luggage to check, you might want to find out if your airline offers an early check-in for luggage the night before. Obviously, this is not worth the hassle if you have to go all the way to Langenhagen for that. But if you are staying in an airport hotel, it might give you an extra hour of sleep in the morning.
Just an idea, not sure if this might work: Have you looked into train connections from Hannover to Amsterdam or Paris? I don't think this will be any more uncomfortable than any of the other solutions, and you would travel with a much better environmental conscience. :-) I hope they will soon reinstall more sleepers and couchettes; there aren't any really good connections from Hannover yet. In case your arrival is via Amsterdam or Paris too, there are some good sleeper connections to Munich.
Celle, Lüneberg, Goslar
Hannover itself can't compete, except for the Herrenhausen Gardens and the Sprengelmuseum (and the zoo, if you like zoos). You might also have a look at Hildesheim with two UNESCO world heritage churches and a renowned Egyptian museum.