What's your question? Ask

Can you tell me a little bit about the climate you live in? What are your summers and winters like? This will help me make a good recommendation for your area. For instance, if you have cold winters and warm summers, Zoysia may work where you are but it goes dormant in the winter and gets brown. If you don't mind trading that for its durability, it would work for you. The most durable cold weather grass is Kentucky Bluegrass. Regardless, you can work to improve your soil by spreading out a little composted manure on the lawn every so often in warm weather. You don't need to work it in. It will work itself into the soil and slowly improve it as you add more.

Conditions such as brown patch can be treated effectively with Banner, Daconil or Bayleton fungicide.

This is something that people do and it does work, but, that being said, fires like this can get out of hand quickly and are often illegal in many suburban and urban areas. If you decide to do this, do it with care and check with local authorities first.

Zoysia grass is a warm season grass and grows well in the south. In your area, you have have the extra benefit of enjoying the grass pretty much year round, as it would only go into dormancy for a brief period of time, if it does at all.

Yes, as long as you live in a suitable location, this dense, warm-season grass may be a good alternative. This article will help you: http://www.lawngrasses.com/dog-traffic/#GRASSES%20FOR%20DOGS

First, I am sorry for your loss. I would have the soil tested to see what nutrients the soil may need and the pH. That being said, a weed and feed fertilizer will help reduce the weeds and give the lawn a good head start. This article will help: https://extension.missouri.edu/publications/DisplayPub.aspx?P=G6706#Maintaining Spot treating with Round-up will help with the weeds. Alternatively, you could try spot treating with natural weed killers, such as vinegar. The following article should be of some help to you: https://www.gardeningknowhow.com/lawn-care/lgen/tips-for-improving-the-lawn-and-reducing-maintenance.htm