Neat! If you have a chance to meet up with Mipa (the author of Alien's Day Out), she is super nice.
There are many Loving Hut branches in Korea and in Seoul. The one in Insadong does all vegan versions of traditional Korean foods, so that's pretty nice. Loving Huts will also be your best bet for vegan restaurants out of Seoul. I know there is one in Mokpo, one in Yeosu, and for sure at least one in Busan.
In Insadong too, there are a couple of traditional restaurants that do (vegan) temple food, which is a very nice experience. You do need to book in advance with them.
Itaewon is the foreigner district in Seoul; if you are craving stuff like hummus or curry that's where you will find it. Close-ish to Itaewon, Haebancheon street is also quite foreign (but more Western) and has lots and lots of restaurant with veg options, including for breakfast. I don't know if giving you any names would be useful, restaurants have a very short life expectancy in Korea so the places I knew might have gone out of business...
Check the vegetarian and vegan Seoul meetup group on Facebook; when I was there we had gatherings about once a month. They also typically have info on their page about bake sales, restaurants, etc.
Other general advice: November starts being quite cold and very dry, so pack accordingly. If you like to hike / walk, Namsan (the mountain in the middle of Seoul) is very nice, especially their botanical garden-ish area (across from the Hyatt). THey have designed it so that something is in bloom all year long. I used to walk Orion there every morning. You can hike all the way from the Hyatt to Seoul tower in about one hour - it's not super well indicated or blazed but basically take any trail that goes up.
For transports, public transportation in Seoul is great, and so are KTX trains to travel across the country. Buy a reloadable transportation pass when you land (in any convenience store) and take the line 1 train downtown, it's just as fast as a taxi, much cheaper and very comfortable. The pass will also get you a discount on all subway and bus rides. The city bus system is a bit of a pain to figure out, but it does reach more areas (no trains to Namsan for instance).
That's all I can think of right now. Any specific questions?