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It can help the grass to retain moisture if you cut it before you take plugs. This will force it to stop respiration and start making new shoots. I would advise mowing before hand.

This could be moles, vols, gophers, or anything that burrows. This article will help you to control this issue: https://www.gardeningknowhow.com/plant-problems/pests/animals/mole-control.htm

As will this one: https://www.gardeningknowhow.com/plant-problems/pests/animals/eliminate-gophers.htm

Slow growth can be common in certain areas. This article will give you information about common issues with this grass: https://www.gardeningknowhow.com/lawn-care/specific/zoysia/facts-about-zoysia-grass-zoysia-grass-problems.htm

I'm attaching an article from the Florida Extension office. It reviews the type of zoysia grasses available and further down covers maintenance, including fertilization. http://edis.ifas.ufl.edu/pdffiles/LH/LH01100.pdf

Yes female dogs and young males that squat to pee will cause brown spots from urine salts. Male dogs usually don't pee in the middle of the grass, but lift their leg against a tree or pole.

Cats usually don't pee much volume so not as much as an issue. And if you put out a sand box or pile of sand the cat will much prefer to use that and not pee in the middle of the grass.

A motion sensor activated sprinkler can deter both.

That is a good solution, and I believe that will work just fine.