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Without knowing your area and climate zone, it would be hard to pinpoint what the actual pest is. This can range from gophers, to moles, deer, hogs. Have you noticed any animals in that area? If they spend much time there, than they are more than likely the cause. You can deter a large number of pests by boiling sliced hot peppers in water for 10 to 20 minutes. Strain this off, and pour the water in the area after the water cools down. This will deter most pests.

This can be common with over-watering. After the grass has recovered from transplant, unless you are experiencing a drought, then it will only be necessary to water if the rains have not been sufficient.

I recommend applying wettable sulfur and dolomitic lime to the area to cure the issue. This will help greatly.

Thanks for the picture. While I favored ground ivy in my previous answer, this picture sure looks like creeping buttercup, Ranunculus repens. It is an invasive plant in my locale but easier to treat than Glechoma. Any broadleaf herbicide meant for grass should work. I've also hand removed small infestations. Quite a bit of foliage comes with each rooted section. The roots are tenacious but short so a trowel or hori-hori knife will pop the plants out. You are likely to have some seed left behind so keep an eye on the area and remove these when they germinate.

This is a common fungal infection. There are many ways to get rid of this from boiling water, to peroxide and water mixed half and half, to wettable sulfur.

I recommend leaving it, however, as it can actually be good for your lawn! Just unsightly. It will clear itself up over time, and even help combat other infections in the yard.

This article will give you more information: https://www.gardeningknowhow.com/lawn-care/lgen/black-substance-on-lawns.htm

If you have zoysia grass, it is probably going dormant. In the north, zoysia is the first grass to go dormant in the fall and the last one to perk up in spring. I think you'll see it recover when temps return to normal.

The hot, dry weather probably killed most of your grass and allowed weeds in:
https://www.gardeningknowhow.com/lawn-care/lgen/reasons-for-dying-grass.htm

Since most of the grass appears to be dead, the best solution is to re-plant the lawn. You'll need to prepare by getting rid of the weeds, whether using mechanical means or a lawn herbicide. In the meantime, make sure the weeds don't spread by mowing them before they get too tall and produce seeds.

These articles will give you some advice on how best to do this:
https://www.gardeningknowhow.com/lawn-care/lgen/overseeding-information.htm
http://www.extension.umn.edu/garden/yard-garden/lawns/lawn-renovation/
http://plantscience.psu.edu/research/centers/turf/extension/factsheets/renovation