Hi there, new FlyerTalk member here. I'm from Hong Kong and recently visited Seoul. Agree with the others that Myeongdong and Hongdae are touristy areas to stay with lots of shopping. I feel like Myeongdong/City Hall area is more of a commercial area, especially during the day so not very interesting to walk around. Itaewon is also an option that is quite active at night, there are a lot of bars and restaurants (both international and Korean cuisine) as well as shopping. Ideally you would want to be situated near a subway station not too far from Seoul station for easy access to the airport.
I personally prefer less touristy and crowded areas, and stayed near Namyeong station which is one stop from Seoul station. There were plenty of restaurant options nearby and it was also not too far a walk to the War Memorial of Korea which I found very interesting. I didn't go into the museum as it was closed at the time I went (past 6pm) but walked around the park surrounding it and there were many military planes, tanks etc on display. Thought you might be interested in that since you're doing the DMZ tour.
As for Hong Kong, since you arrive late you might want to stay somewhere that is convenient to travel to from the airport - Central, Admiralty, Tsim Sha Tsui (TST) would be good areas depending on your budget. You only have the morning and early afternoon to explore, and I would recommend taking the Star Ferry across Victoria Harbour between Central and TST for a beautiful view of the city. Ticket fare is very cheap.
If you're into shopping and street food you could head to the Ladies' Market in Mong Kok where you'll find a variety of stuff (not just for ladies, and remember to bargain) as well as reasonably prices local restaurants and snacks on the street. For higher end shopping there are plenty of shopping malls in Central, TST, Causeway Bay.
The airport express runs between HKIA and Hong Kong station and takes 23 minutes, and is the fastest way to get to the airport. There are also busses throughout the city that go to the airport which are a cheaper but slightly more time consuming option.