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For best results, soil testing is recommended before fertilizing. A soil test will indicate major nutrient deficiencies and the pH of the soil. Slightly acidic soil (6.0-6.5) is best. Lime should be applied only if the pH is less than 6.0.

For routine maintenance where soil tests indicate no major deficiencies, use a lawn fertilizer with approximate nitrogen (N): phosphorus (P): potassium (K) ratio of 3:1:1 or 4:1:1 or 4:1:2. For example, a 16-4-8 fertilizer has a 4:1:2 NPK ratio.

Where soil test indicates low phosphorus or potassium levels or where basic fertility levels are not known, use a fertilizer with a ratio that more closely approximates 1:1:1 or 2:1:1.

Here is an article that you may find helpful: https://www.gardeningknowhow.com/lawn-care/specific/zoysia/zoysia-diseases-tips-for-dealing-with-zoysia-grass-problems.htm

humm you might have grubs!!! Do a quick test on an area, dig and check to see grubs!! if you find more than 2 in that spot even one for me, you have grubs!! If you saw mushrooms growing in that spot... then you do....

I use garlic for the fungus, its good, you dont contaminate, but you have to apply it every 7 days. I also use NEEM OIL this stuff is AWESOME goes for everything, the only down size of this is you have to be consistent with it.
You can spray everything with this, plants and grass!!! Good Luck

Generally, Zoysia grass should be kept at a mowing height between 3/4 to 2 inches.

This article should help:
https://www.gardeningknowhow.com/lawn-care/lgen/fill-uneven-lawn-low-spots-how-to-level-a-lawn.htm

It sounds like you have leaf spot. This article will help:
http://www.aces.edu/pubs/docs/A/ANR-0621/ANR-0621.pdf