Gazoinks: Ouch, that's pretty nasty. We had an 11 hour layover in South Korea when we went to China, but we ended up just leaving the airport and taking tours and such.
StingingVelvet: Here in Tbilisi there is no real public transportation to make that affordable. I have hard Istanbul, where the second layover is, has a nice metro system from the airport to downtown. Still, with customs and visa procurement and travel time and security I don't think 7 hours is long enough to do it all comfortably.
Maybe, we shall see.
I can't speak about Istanbul, but it wasn't too much issue getting out of the airport in Korea, as I recall.
May I suggest building a hand-cranking mechanism to charge your laptop? :)