Are there one or more play areas? There is one. It's big and centrally located.
Is there one in every terminal, after passport control? no-- the two satellites have no play areas.
Is the play area conveniently located? yes, in a sense that it is near the major restaurants. However, the location is not fantastic in terms of getting to gates for departure-- you have to allow a good 20 minutes for walking to the gates that are located in the concourses, and quite a bit more for taking the train to the satellites.
Is there information about the play areas available online? yes, it is on the interactive airport map at http://www.portseattle.org/about/maps/cte.shtml
Are maps clearly labeled to make finding the play areas easy? yes-- there's a teddy bear symbol. It is very close to the central food court area.
Are there one or more nursing areas? yes, as well as a family restroom, attached to the play area.
Description: The play area is relatively clean, though it was beginning to show wear. I never saw less than 10 kids running around there, so it must see some pretty heavy traffic. That, combined with the fact that it is quite a hike to most of the gates, leads me to conclude that Sea-Tac needs to sprinkle a few smaller play areas around the remainder of the airport. The family restroom is a nice touch, as is the fact that the play area is without a television. Kids traveling on airplanes are already overstimulated, they don't need cartoons running in the background, too!
Note for military families: If you have tons of time, enough that you could leave the secure area, come back in and still make your flight, then the Sea-Tac USO is small but fantastic. Loads of free snacks (including milk and sandwiches, healthy kid options!) and a small playroom are available there, in addition to a tv room.