Hey everybody, I'm back with more oh-so-fascinating photos.
So, an update.... its the middle of July, hot as Hades, and I'm happy to report that the Zoysia is, for the most part, in great shape and growing well. I bought a core plugger/aerator and loosened up some of the compacted areas, fertilized a couple times, applied some fungicide, pulled weeds by the hundreds, and I water a couple times a week (otherwise some turf gets drought stress).
All in all I am pleased. The areas that get full sun are magnificent and lush, like astroturf. I found out that my zoysia is "Compadre", which I think is OK. The areas that get a lot of shade look pretty pathetic... but no sun = no zoysia, it seems. I'm mowing the grass at 2" now, since it seems to do better in the heat, and thus requires less water.
But.... I've got a few patches of strange-looking zoysia, in among the otherwise healthy turf. The blades look withered, a tad bit yellow, and seem to be more "stem-like" since the blades are curling inwards on one side. It definitely does not seem like a drought response, since these strange patches have been growing slowly larger for a couple months. I figured they would go away, but they are getting bigger.
Many thanks to all of you for your help.