Agree with this. Asiana is a very good airline (similar with Korean Air). You can get a room at the airport only for the day. Also there are places to get shower and massage for reasonable price. The airport is the best (IMO based on the airports I have seen including HK) I have seen. Everything will start opening up around 6-7AM. When you first get there it will be very quite but will soon be busy. If you are flying Asiana all the way back to USA and have your tickets for the SK-USA flight and baggage checked to USA then you do not need to go through customs. You can go through the transit screening line by-passing customs. If you need to check baggage/get tickets then yes you need to go through customs. They also have some 4-6 hr day trips around Seoul that are available, as I understand it, to all transit passengers. You can check this out on the airport website.
