Hi!
I have been to Korea more often than I can count probably, and I thought I'd give you my opinion on the Seoul part. Please find my comments in color below:
Day 5 - Travel/Seoul- Take the train from Dongdaegu Station, Daegu to Seoul (app 2 hrs journey time).
- Check-in to hotel, spend time exploring Myeong-dong neighborhood.
- Overnight: Seoul 1/6 (Myeong-dong) Good choice of neighborhood to stay, although not the best for nightlife. Myeongdong is definitely the main shopping area, and it's also nice to test out all the streetfood! Also, if you feel like it in the late afternoon / evening, take the cable car up to Namsan tower. It's a super nice view on the city, usually an activity for couples. If you feel super fancy there is an expensive but good restaurant on the top floor of Namsan tower - probably number one spot for proposals in Korea.
Day 6 - Seoul/Day Trip to DMZ- Book the "DMZ/JSA & 3rd Tunnel Tour (Seoul)” with Koridoor Tours (we’ve read good things about them here in the forum).good experience
- Overnight: Seoul 2/6 (Myeong-dong)
Day 7 - Seoul- Visit Insa-dong, Samcheong-dong, Namdaemun Market On this day, to cover what your plan was, I would probably plan a route to walk from Myeongdong to Gyeongbokgung Palace, then a bit north-east into Bukchon Hanok Village, then back down with a stroll along the Cheongyecheon (a cute stream flowing through the center of Seoul which is nice to look at, especially in spring) and back to Myeongdong. Namdaemun is more of a 1 hour walk-through and nice to see, but there is not that much to do unless you want to buy some fake purses or fabrics and eat streetfood (which you can pretty much find anywhere anyway) - would definitely visit, but not plan too much time for this.
- Overnight: Seoul 3/6 (Myeong-dong)
Day 8 - Seoul- Switch hotels from Myeong-dong to Gangnam (just for a little variety/change of pace?) - I 1000% do not recommend switching hotels to Gangnam. It's like moving from the heart of the city to the boring office district. There are some fancy high rises and some cool stores e.g. Samsung d'light and a lot of expensive designer chains especially in Cheongdamdong, but it is not vibrant or exciting at all in my opinion. I would probably take a stroll there if I was you for a couple of hours and then move on to another activity / area. There's so much more to see! If I can suggest something else: Move your hotel to Hongdae! It's the "student" area of Seoul with exciting night life but also cool, unconventional shops, some funny museums (Trick Art Museum comes to my mind) and good food.
- Explore Gangnam neighborhood.
- Overnight: Seoul 4/6 (Gangnam)
Day 9 - Seoul- Visit Lotte World - medium recommendation from me. I've been there twice and it's cool if you're living in Seoul and literally have too much time and money, but the queues are endless and the attractions are mediocre in comparison to six flags / disneyland / any European amusement parks. The shopping mall around Lotte World is quite cool, and there is a nice aquarium in that complex as well, and during winter a nice ice rink, but I would not use an entire day of your holiday for this. Alternative recommendation: Do half a day of spa e.g. in Dragon Hill Spa - better value for money imho
If it HAS to be an amusement park I would recommend Everland instead. But the queues there are also pretty unbearable, especially if the weather is good.
- Spend more time in Gangnam
- Overnight: Seoul 5/6 (Gangnam)
Day 10 - Korean Folk Village- Day trip to Korean Folk Village? - I have actually never been there but it seems quite far away. Go for it, or consider any of these areas to check out: 1) Dongdaemun - the fast fashion monopoly. Huuuuuge buildings top to bottom filled with clothes (it's basically a wholesaler) 2) Itaewon - this is the foreigner district and there's not that much to see as a tourist, but Vatos is a mexican restaurant that sells mexican-korean fusion food and is really good (Kimchi fries is all I am craving sometimes). You could also visit the war museum in that area, which is super interesting. 3) Noryangjin Fish market
- Overnight: Seoul 6/6 (Gangnam)