Singapore Air is great to get to Tokyo, same with JAL. Stay away from United. If Seoul, fly Korean Air. They have a non-stop from Dallas and one from Houston...great flight! China Air is fine getting into and around China, from either Seoul or Tokyo. Chinese airports are a pain but certainly doable. We moved around in China with kids...and made it.
Haven't spent much time around Japan other than transfers and a couple of days here and there. Others seem to have a few good ideas on rail passes, etc.
Seoul is a BIG city. Frankly, not much to do that is different from any other big city. There is a ski resort a little south of the city. Other than that, the nightlife is pretty good and typical Asian. I would suggest going to Busan instead. Plenty of temples and sights to see outside the city and the city itself is fun - a lot of expat nightlife, a beach and plenty of good eats.
Bejing is awesome. I would spend most of my time there. So much to see and most is walking distance from Tiananmen Square. Markets are a good time, some of the best restaurants around, temples and museums, on and on. The Great Wall is unbelievable. Make sure and have your hotel set up a reputable tour, and try to get a private one. You will have most of the day at the GW. They will take you to a touristy-type place - either clothing or pottery or something like that, usually something a relative owns. No pressure to buy just part of the act and frankly, not too bad. Also, make sure and take the toboggan down from the wall after. I'll go back and try to find out but there is a preferred entrance (several to choose from). It is the one with the Subway shop there. I believe the only one with the toboggan run, too.
Also, from Bejing, you need to get to Xi'An and go see the Terracotta soldiers. That is a day well spent and a true wonder. The city of Xi'An is not too bad either, plenty of great restaurants though were happy with the hotel ones.