Where to stay in Venice?

I'd suggest the Park Hotel Annia. www.anniaparkhotel.it It is very comfortable with a huge number of amenities. It is a mile from the Marco Polo Airport and has a free shuttle to the airport that starts at 4:30 am (and leaves every 30 minutes) and gets you to the airport in 3 minutes so no taxi needed. To get to Venice for sightseeing, just take a water taxi. Their wonderful breakfast buffet begins at 4:30 am so you can fuel up before taking the shuttle. Since the shuttle will get you there in 3 minutes, you can leave later than you thought. The rooms are soundproofed so you'll get a quiet rest. We like their suites (very reasonable price) but there are rooms for every need at great prices.

I wouldn't stay at the airport, as suggested above, except for maybe the last night. Staying near the airport for your entire stay would make it very inconvenient to visit the city and very expensive. The water taxi from the airport costs over 100€ a ride.
A better option, in my opinion, is to stay within walking distance of Piazzale Roma, where you can catch a taxi (one with wheels, not a water taxi) or even the bus. For a 7am flight, which is a intra European flight, you would need to be at the airport before 6am. The taxi ride from Piazzale Roma is about 25 min, the ATVO bus takes about 35 min.

I don't know what your budget is, but the hotels below are nice and only 5 minutes walk from Piazzale Roma. If they are not to your liking, or not within your budget, there are many others in the area.
http://www.mgallery.com/gb/hotel-1313-hotel-papadopoli-venezia-mgallery-collection/index.shtml
http://www.hotel-olimpia.com

Another observation. If you are staying at the Cinque Terre prior to Venice, it might be a good idea to return the car in La Spezia and go to Venice by train. You won't need a car at the Cinque Terre.

The No 5 bus goes from Piazzale Roma at 4:35 and gets to the airport at 4:58. 8 euro per person.