There are two storage services groundside that I'm aware of, and both will let you store your luggage for a reasonable price. I helped a friend store a lot of his stuff for a week as he came by for a visit on the way back to Europe from an extended stay in Australia and didn't want to lug all his stuff into Seoul. We ended up using the one that isn't the post and it all worked pretty easy. I think the post one was cheaper, but it closes quite early and was already closing up as we arrived around 8 or 9 PM, if memory serves. I'd point you to the website, but it seems you already found it and for some reason it won't load for me right now. I think somewhere on the website it says the opening hours for both.

As for checking in with the airline already, I know people who did that in various German airports, but that was a while ago and I haven't heard of anyone doing it in ICN, or even in Germany recently.