Answer: Based on your description I think you may be facing several problems in establishing the lawn. Zoysia requires full sun all day to stay healthy, so it will not thrive in shady areas. The other more shade tolerant lawn grasses still need at a minimum a half a day of sun to grow well. So if you have somewhat shady areas you may need to consider establishing a shade tolerant ground cover in those areas instead of grass. Lawn grasses are not able to compete with tree roots, so if you have extensive areas near trees, this may also suggest using a groundcover in those areas.
In general, a healthy zoysia lawn will crowd out weeds. Zoysia requires a different regime of care from the other types of lawn grass. Here is some information about growing zoysia -- a warm season grass -- and about growing cool season grasses. It was developed for Virginia and Maryland, however should apply in your area as well. YOu may need to cut and paste the complete urls to make them work correctly.
I hope this helps!
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