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It could be grubs and it also could be a fungus. Zoysia is normally pretty resistant to both, but if it has had a hard season (which may be possible with the weather), it would have been more susceptible to these problems.

These articles will help you identify which it is and how to treat it:
https://www.gardeningknowhow.com/lawn-care/lgen/grass-fungus.htm
https://www.gardeningknowhow.com/plant-problems/pests/lawn-grubs-how-to-get-rid-of-grub-worms.htm

If you are dealing with partial shade, I would recommend the St. Augustine grass. It is better for both warm climates and shady conditions.

It is likely dormant. Many lawn grasses will go dormant in dry weather to preserve themselves, so this is not unusual. It should come back just fine in the spring when the temps and water return at the levels it likes.

Zoysia grass is a very invasive grass. It can be difficult to control. Small areas of Zoysia can be treated with a non-selective type herbicide, such as Round-up. However, this type of herbicide can potentially kill any plant it comes in contact with, so be careful when spraying around plants you want to keep!

For best results, apply the herbicide when the Zoysia grass is still green, actively growing and not dormant or under drought stress. Since Zoysia grass may regrow, more than one application will probably be needed. Since Round-up dissipates rather quickly, it shouldn't pose any problems with future planting. This article may also help: https://www.gardeningknowhow.com/lawn-care/specific/zoysia/controlling-zoysia-grass.htm

Larger lawn areas can be controlled by putting a tarp over it at any time, even after the grass has started to come out of dormancy. The lack of light will kill it.

Pet urine, especially from dogs, is pretty toxic to almost all plants (including zoysia), especially if it is done in the same spot multiple times. Most of the time, people will either teach their pets to go in one specific area, so the spotting is contained or will dilute the spot after the animal has urinated. For additional help with treating urine spots in the lawn, this article will help: https://www.gardeningknowhow.com/lawn-care/lgen/dog-urine-grass.htm