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Seeding may help. I think that until the mole cricket problem is fully addressed, doing anything more than seeding would just be a waste of money and time as the area would just be attacked again.

Yes, zoysia will grow in your state. In fact, there are many varieties of zoysia that have been found to thrive in northern climates, especially those that have warmer summers. A few that come to mind include Zenith, Japonica, and Amazoy (also known as Meyer).

This condition with seeds is called helmet head. It is when the seedling does not discard the seed coat as it should. Normally, it is caused by too little humidity but there are several things that can cause it. It will correct itself, but watering can help speed it along.

Burning to remove thatch can be done in the early spring. Check with your local government on regulations in your area.

Zoysia is extremely difficult to get rid of. You can either mow it down as close as possible to the ground and treat with herbicide or remove the sod with a sod cutter. Small areas or patches of Zoysia in a lawn can be treated with a non-selective type herbicide, such as Roundup. However, this type of herbicide can potentially kill any plant it comes in contact with. So be careful when spraying around plants you want to keep! For best results, apply the herbicide when the Zoysia grass is still green, actively growing and not dormant or under drought stress. Since Zoysia grass may regrow, more than one application will probably be needed. Here is additional information: https://www.gardeningknowhow.com/lawn-care/specific/zoysia/controlling-zoysia-grass.htm

Did it die when the cold weather came? Zoysia grass goes dormant when it gets cold and with the odd weather this year, many places that do not get cold enough for it to go dormant. If this is the case, it will green back up over the spring. If it died before the cold, please let us know the pattern it died in (i.e. all at once, in circles, in patches) and we can better help you with this.