Some advice from a Norwegian:
Park Inn and Radisson Blu are both at the airport proper. I usually stay at Park Inn (turn left, five minutes walk), as it is cheaper than Radisson Blu (turn right two minutes walk). Both are overpriced for dining.

If at all possible, do NOT take the Flytoget express train, take the local or regional train, as it is half price (NOK 92 as opposed to NOK 180 O/W), and only five minutes slower. Three trains every hour - a rather similar situation to the Heathrow Express/Connect set up.

There is a 24 h ticket available (24 h from validation) at 190 NOK, which also is valid as a travel card on all other public transportation within Oslo as well as the local/regional train to/from the airport (NOT on the Airport Express Train).

Personally, with a 07.25 departure I would stay both nights in Oslo proper, preferably at one of the hotels in/very close to Oslo S. Comfort Grand Central, Scandic Byporten and Thon Hotel Opera are in the station building/on railway property and are quite OK.

There are several others literally across the street, including Radisson Oslo Plaza and Clarion Royal Christiania mentioned by the previous poster. There are also hotels next to Nationaltheateret station, where all local/regional trains to the airport stop, as well as half the Airport Express Trains. This is a much nicer area than the area around the main station (Oslo S)