Agree with above that knowing what variety you have would be more helpful, even just the main type of if it's a japonica or matrella.
That said, mowing height is like most other warm season turfgrasses: you can mow it higher if you want to water less and let it's roots develop more depth or mow it lower for a cleaner look, but you'll need to water more frequently.
Zoysia is more drought tolerant than St Augustine or Bermuda so I'd think you'd be ok with 30 minutes of watering a week with the temps we have right now. Are you watering in the early morning? Watering mid day is very inefficient and you lose a lot to evaporation. Early evening can be bad as you don't want a lot of water not being absorbed and getting fungal problems so make sure you're watering in the morning.