Should I also lay down some seed to try to fill in any spots occupied by weeds?

Hi. Zoysia is one of the first grasses to curl lengthwise (like a straw) to conserve moisture in droughts with heat. That's why it survives drought so well. It takes protective action faster that other grasses. In the extreme rainless drought we've had this year that hasn't been seen like this for 76 years, my zoysia began curling much earlier than my bluegrass--many, many days earlier--a week or two earlier, I guess it was. It was something to see some of my backyard zoysia look like fine fescue (I keep my front yard zoysia watered more than the back).

Like most grasses in the summer, zoysia needs about 1 inch of rain or sprinkler water a week to keep the curling and browning away. Zoysia can suffice with a little less: 0.75 inch a week. Lay out tuna cans or regular-height tin food cans to measure how much sprinkler water your grass is getting. The cans can also measure rainfall too.

I wouldn't be cutting your zoysia lower at this time, with conditions still so hot and dry. A little higher might help right now. During the spring I was cutting my zoysia down to a height of 2 inches every week, but during this very unusual summer we've been having, I've been cutting it down to 2.6 inches, and haven't needed to cut it weekly or even every other week. My front yard zoysia is green right now and been getting compliments. But watering is key along with cutting less frequently during this crazy summer that hasn't been seen in the U.S. in 76 years.

About the fertilizer you mentioned, don't fertilize zoysia this late in the season; it could harm the zoysia this late. There are other threads here about when to fertilize zoysia. Next year is your safest bet now.

Looks like drought and heat stress to me. That being said last thing you would do is apply fertilizer, scalp it, or dethatch it unless you want to kill it off. Sort of like a boxing match and you knock your opponent to the floor, then shoot him in the head with a gun to make him get up and fight more.

What have you done to water the grass?

Agree with TW. What was your watering schedule? How often and for how long?

At this time of the year, the only thing you want to be doing on zoysiagrass is mowing as needed, and applying the correct amount of water at the correct intervals. As the days continue to get shorter, the warm-season grasses such as zoysiagrass are slowing down their growth and development for the season.

As was previously mentioned, this is the time of the year when you would want to start raising the cutting height on warm-season grasses such as zoysiagrass.

Do NOT cut it really low; you can bag the clippings; Do NOT apply any fertilizer or herbicide!!!; Do NOT dethatch now.

Have you ever taken soil samples and submitted them to a soil lab for testing?