My guess is that it should be checked all the way through unless you ask them not to.

That is a long layover. Getting into the city is pretty simple so I would probably do that for a while. Seoul is a "city that never sleeps". But it sort of depends on what you like to do. I just spent a week and there were some interesting things to photograph at night which is something I enjoy. Depending when in March you are there it might be pretty chilly. In early April it was quite cool at night. Seoul has a very good tourism website so you can check that out to see what there is to do.

If it was me I would want a room to relax in. But I also like my sleep. There's a Hyatt right by the airport that I think you can get back and forth from on a maglev train. But staying in the city is also easy enough. You'd just need to be up early enough to get a train around 7 to get back to the airport.