oh gad, in Taiwan for 4 days and you spend it all in Taipei. Taipei 101, Sun Yatsen Museum, National Museum, night
markets, maybe a hike to the hot pools and a soak.
There's a suggestion to go to Din Taifeng. Bleh, there is so much good food and dumplings in the small shops all over the city. Din Taifeng is good, don't get me wrong, but it's hardly 'an experience'. It's like going to PF Changs.
Just go out on foot and ask the locals for SHWEI-JYAO and you'll get dumplings.
If it were me, I'd spend two days down south in Kenting. The beach, the mountains, the Taiwan culture that isn't CITY culture. Hengchun and the Kenting night market are great fun.
Korea - very hard to 'see Seoul', about the only thing you could really do is get into the city and maybe hike up tot Seoul tower. Incheon is about an hour out.
I'm no help on the other two, only been to Japan for work. Never been to Vietnam.