South Korea has very strict quarantine requirements for dogs. If you meet those, you should be fine to leave the airport with your dog, but they're not the easiest requirements. I haven't seen dogs off-leash on the air side at ICN in all our trips with our dog there, but I have seen several after passing through arrivals.

We've had a tough time finding hotels that will accept our dog, too, as most we have contacted have strict 5-kg limits, but there are a few just a few minutes from the airport that we've stayed at in the past.