1. I've never taken the ICN-offered tour before, but what I heard from people that have is that the tour very briefly touches a few spots in Seoul. Also, if you have a longer layover, it doesn't really matter as the tour gets you back to ICN very early. What would I recommend? Let me work backwards. Assuming a 6pm departure, then roughly a 5:30pm boarding time. If you arrived back at the airport by 4:30pm, that should be plenty of time to make it to the gate. For me, I would get back a bit earlier (4pm?) and grab a quick shower in the lounge before getting on a long haul.
What to do? Lot of choices. Something cultural?
Gyeongbokgung Palace is a great place to visit. They have a
changing of the guard ceremony which is fantastic to watch. The palace is closed on Tuesdays. But there are other palaces to visit if you are arriving on a Tuesday.
Then after visiting the palace, if you head to the East, you can see the
Bukchon Hanok Village, a traditional Korean village. It is not a tourist attraction, but an actual residential village with most homes there built traditionally. Unfortunately the number of tourists visiting there have skyrockets in the last few years and residents have started to complain. The complaints seem to be geared towards large tourists groups led with megaphones and being disruptive. If it is just you and one other, I'm sure you will be fine.
Lastly,
Insadong is a great place for visitors. It is a great place for souvenir shopping.
2. I am not sure if you can bail out on the tour midway, but given the above, I think it should be enough to do.
There is a place in ICN airport that has luggage storage service. Check
here. Also there is a luggage storage service
at Seoul Train Station. But I figure it would be easier to leave it at the airport.
Getting into Seoul is easy. Let's assume you have cleared immigration by 7:00am. From here you have two good choices for getting into Seoul. The Airport Train or the Airport Limo buses. The Airport Train (AREX) is made up of two different services. The non-stop Airport Express train goes from ICN directly to Seoul Station.
More info here. Sometimes if the Express train has just left, the staff may direct you over to the Commuter All-Stop train. The
All-Stop Train takes 15 minutes longer, however it beats having to wait 30 minutes for the next Express train. Once arrived, take the multiple escalators up to street level. From there, take a taxi to your destination.
The Airport Limo bus is also an excellent choice for getting into Seoul. It won't take you to a palace but will get you to one of the hotels nearby. You can get a feel of the city just looking out the windows of the bus. But the downside is that you might get stuck in the morning rush hour traffic as you get closer to the city.
Let me know if you need any more information. Monsoon season is starting soon so hopefully you arrive after that.