"(where do I leave my backpack?),"
There are public lockers at Seoul Station but I'm not sure if you can leave stuff in them for a week or more? It's also about 4000 won ($3) a day so 9 days would be $27. I'd cut one day from Seoul & spend it hiking in Seoraksan, on the 24th October, & get a flight from Jeju to Seoul on Saturday October 29 th instead. If you're staying at a guesthouse in Seoul maybe you could drop in on Oct 22 nd & leave it there for the 9 days. They usually just ask you to fill out a name tag, sometimes in duplicate. You will have more chance of doing this if you've paid for the room upfront. Btw there's nothing much to see at Jeju City, & the bus terminal to get out is located about 1 km inland from the port. I started walking left along the road following the shoreline & only found a taxi rank & tourist info office. (Taxis in Korea are cheap, btw, unless you're traveling long distances, when it can add up to over 100k won) Seoul Airport to Seoul City is 80k won, about $70. Cheaper to get the airport train for 4k 'all stations', or 13k express. There's only 10 mins difference in the time between the 2 trains, so the all stations is fine. Just outside Seoul station is about the only place in Korea where you will see beggars. They can be a nuisance, but are harmless.
