I have a few hours to kill at ICN. Is there a gym at the airport? Would going to the Hyatt to use their gym be a good option? Could I go for a 40-minute run in the terminal and then shower in the CX lounge (I ask this only half facetiously!)?

I don't believe there's a gym at INC, but there is a spa with massage service. There is also the Haesoopia Spa complex nearby with hot salt pools and all the services that go with that kind of facility, but no gym:

http://www.haesoopia.co.kr/

Haesoopia offers saunas and hot springs using deep-sea water. It has various facilities including mugwort baths, charcoal saunas, jade saunas, open-air hot spring baths and open-air valley baths. The most popular are the open-air hot spring baths and valley baths.

Deep sea water extracted from the rocky ocean floor has a much higher level of essential amino acids like calcium, potassium, magnesium, and sodium that are essential for the human body. Taking a bath in deep-sea water generates osmotic pressure caused by the difference in salt concentration between the body and the seawater, thereby facilitating the elimination of waste products from the body and activating the metabolism and the circulation of blood.