AFC is pretty clearcut - you can't argue any of your picks. The Colts are probably the best regular season team in the AFC, and the Steelers and Pats obviously found plenty of success.
The NFC is a mess...
I think the Eagles have to be #1. I would wager they have to have the best record in the NFC in that timespan. If not the best, definitely close. They also only missed the playoffs 2 years, and made the NFC championship game 5 of the 9 years. They also made it to at least the divisional round in each playoff appearance, and never were 1 and dones.
I suppose the Giants make a convincing number 2, carried mostly by their two super bowl appearances and 1 victory. They made the playoffs 6 times this decade, but they were 1 and done 4 of those appearances. The other two went for a super bowl run. However, they weren't the team to beat in the NFC until they won the Super Bowl - the Eagles had that title for a majority of the decade.
Third place... eh, Seahawks make a good case, but still somewhat flimsy because of their weak division, and the fact that they weren't even in the NFC the entire decade. However, due to a lack of anyone better, I guess you're right and the Seahawks make the cut. They have a Super Bowl appearance and a lot of seasons winning the division... if they were excluded because of their AFC roots, I think I'd go with the Rams - they won the division for nearly 1/3 of the decade, and had a Super Bowl appearance... in all reality, they did just about everything the Seahawks did, and they've been in the conference the whole time. So I edit that former statement, and now pick the Rams ahead of the Seahawks.