Seoul

I can answer question 2: Yes, the train from Incheon Airport to Seoul Station (downtown) has been operating since late last year. It is called AREX, and you can find full info here:
http://www.arex.or.kr/jsp/eng/main.jsp

thank you very much, that is very helpfuk.

you can check transit tours in below web sites. There're many options.

www.hanatourintl.com

hi. i can't speak english well. so please understand my poor english.
there are many beautiful place in korea.
but i'm sorry that u stay 1.5days only in seoul.
the weather forecast predict that it'll rain a lot this weekend(it's also going to rain now). don't forget an umbrella.

many treavelers usually visit insa-dong. because they can enjoy many tradional thing(?).
if u interestd in shopping, u should go dong-dae mon. there are many chip and high value stuff.
if u interested in baseball, u should go jamsil baseball stadium. we have very unique chanting culture dislike other country.
i'd like to tell u about seoul more detail. but it's time to go to bed.(actually it's too hard to write in english for me)
so i will reply again in the evening.

ps. after visiting korea where will u go?

Actually, the latest weather report has a bit of clearing forecasted for Saturday and Sunday, so you may be in luck! But it is rainy season, so make sure you have an umbrella handy.

If the weather's clear, I'd suggest Insadong (though highly touristed), Gwanghwamun Plaza, and Gyeongbok Palace. They are all within walking distance of each other... You could take the metro line #3 (orange) to Anguk Station (안국역) exit #6, walk along Insadong (stop at Jogyesa Temple, a side-street over), and then walk down to Jongno. You can also see Tapgol Park (to your left) or Bosingak (to your right). You'd want to head towards Bosingak to get to Gwanghwamun Plaza, though. At Gwanghwamun Plaza, you can see the statues of Yi Sun Shin and King Sejong the Great. There is an intersresting (and free!) museum about hangeul, the life of Sejong, and Admiral Yi Sun Shin that can be entered at the rear of the Sejong statue. If you keep walking along the plaza towards Gwanghwamun Gate, you can enter Gyeongbok Palace. Admittance is just 3,000KRW and includes admission to the National Folk Museum and the National Palace Museum.

If it's raining, you could go to the National Museum of Korea... it's wonderful! You would just need to take the metro to Icheon Station (이천역), and the exit is clearly marked. Admission is free, excluding any special exhibits.

For information on the subway, check out this site.

One correction on above.

National Museum is in Ichon (이촌역), not Icheon (이천). Icheon (이천) is near Yongin and Yeoju. Some names are very similar.

Ichon, right... I started typing Incheon out of habit. ㅠㅠ

I've written a few times about things to do in Seoul.
http://www.aussieontheroad.com/sightseeing-in-seoul/
http://www.aussieontheroad.com/vital-visits-in-itaewon/

The former is more touristy. The latter is probably better suited to expats.

As above, the subway does run, but I would say it's a long boring ride with nothing to see. The limousine bus is a little more expensive, but runs every 15 minutes or so, and there are lots of them going to all different parts of the city, and at least you get a view all the way.