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You may have slime mold. This article will help: https://www.gardeningknowhow.com/lawn-care/lgen/black-substance-on-lawns.htm

Zoysia has quite a few weaknesses as a pasture plant. While, it may be tough enough to tolerate harsh conditions and heavy traffic, making it good for lawns, it isn't very tasty or good for horses. The grass produces mildly toxic substances, which are meant to repel pests, but can irritate horses. Therefore, it may be best to steer clear of zoysia grass for use in a pasture.

Zoysia is pretty hardy. It should not affect the lawn at all.

Zoysia is invasive but in a yard like yours it would be a good option. It won't kill plants larger than itself, so plants like hostas and the like will be fine. It does like to get into flower beds and ground covers, so you would need to make sure you had borders around those places and edge them yearly to keep it from getting in.

Really, any grass that is good for large areas because it spreads well has this same issue. They spread well and stay lush because they are aggressive plants. Bermuda grass is another good option.

I am not sure if they are new, but take a look at Zeon and El Toro zoysia. They are considered the most shade tolerant.

This is standard behavior for zoysia and is the reason it is not used much in areas that get cold. The brown patches are where the grass is going dormant when the weather starts to get colder. The green grass is likely not zoysia, but another type of grass. Sometimes, in colder areas, zoysia is mixed with a grass that does not go dormant so that the lawn still looks nice in the fall and winter. But, over time, the two grasses compete and push each other out, so in some areas you will get patches of the non-dormant grass and in others patches of zoysia.

You can try overseeding with another kind of grass in the spring, but eventually the two grasses will go back to patches. Other than that, there is no way to keep the zoysia from browning in the fall if you live somewhere where it gets cold.