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This can happen in colder regions. I would not do so at this time. You will want to let it establish before you use anything that might inhibit rooting. Here is more information on the grass: https://www.gardeningknowhow.com/lawn-care/specific/zoysia/facts-about-zoysia-grass-zoysia-grass-problems.htm

Sure. There would be no harm in this. This article will help: https://www.gardeningknowhow.com/lawn-care/lgen/overseeding-information.htm

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These are the guidelines that I was able to find.
These are only guidelines and I cannot say for sure that you would not have damage to the grass.

You will need to test the water. The chlorine level should be no higher then 0.1ppm.
PH should be in the neutral range 6.5 to 7.8.

I also should point out that the research showed that most cities have strict guidelines on discharging pool water in the sewer system and you may need to notify the city.

I would wait until it is actively growing and you are past your frost-free date. Blades that are still dormant will not absorb nitrogen and it is the one nutrient that leaches through the soil the fastest. Tender blades responding to increased nitrogen will suffer the most if you have a freeze later.

Can you include photos, and your climate zone? This can help me to better determine what is going on. This can be anything from animal urine to soil infections. Has it been overly wet lately? This can cause fungal infections, which can ultimately lead to the death of the yard. Putting down wettable sulfur and dolomitic lime every year will help combat this, and prevent future issues.

This link will help you with maintenance scheduling.
https://hgic.clemson.edu/factsheet/zoysiagrass-maintenance-calendar/