Yes. It is dormant and there is no issue with laying sod in the winter.
The key is NOT to over water your new sod. Watering requirements depend on the weather and time of year you lay sod. For example, in the Spring, you may need to water an hour every other day. In the heat of the summer, you may need to water for an hour each day. It all depends on the humidity and heat of the year.
Yes, you can. The majority of zoysia sold is for areas where Bermuda cannot grow due to sunlight restrictions. Zoysia can be planted in full sun areas as long as you have a proper irrigation system.