Although there's various
gardens inside the terminal, these are not usable for pet relief purposes
. There is no pet relief area available to transit passengers inside security at Incheon. You and the dog will need to pass immigration and exit the terminal order to give the animal a potty walk during the layover. ICN (rightfully) boasts about being a very efficient airport, so the time you'll have will be more than sufficient for this.
There's grassy, tree-lined patches directly next to the terminal buildings separating the terminal from parking areas. Best in area "C".
However, there's a new terminal currently under construction where Korean is supposed to move once it is completed later this year so things may well be change until your flight, especially when booking KE.
As far as selecting airlines goes, the main difference between Korean and Asiana is their seating layout. Eschewing the prevailing trend towards 1-2-1 seating that Asiana (like most other airlines) has adopted for their newer premium cabins, Korean opted to install seat pairs (2-2-2) in even their newest configurations. This is better if you travel with a companion. You can put the carrier with the dog between the seats.
Traveling on your own, Asiana is the better choice. Select a window seat with the seat facing the aisle. You can place the carrier between the window and your seat - very private, and always within reach, even when in lie-flat mode.