They can and will check it onwards - it often goes on the same bus as you, I've seen my bags get loaded into the same bus I've on many times. You will see someone at the baggage carousel sitting watching for tagged bags.

Practically, they will also do this if it's not on the same ticket - they are very cooperative and helpful. They'll also print your boarding passes at PUS/wherever you are as well even if not on the same ticket. You can also buy the bus tickets at GMP/other Korean airports. If you get a jet bridge, from stepping off the plane if your timing is good you can be on the road in <5 minutes.

Two hours is tightish but I've done it many times. In fact I've done it the other way around including clearing immigration, customs and made it without a problem. I'd book more if you don't know your way around it, but I wouldn't be *that* worried. There are so many flights between PUS and GMP that you should be able to get an earlier one unless you are going for a early flight out of ICN. In which case you should enjoy the Grand Hyatt, it's very nice.