A pre-emergent, probably mid-march depending on what part of GA you are in. I would recommend Dimension or Barricade which you won't find in the big box stores. If your limited to the big box stores, look for Scott's Halts As far as the fertilizer, it depends on whether you want to go organic or synthetic. If you elect synthetic you want to wait until the lawn starts greening. Organic doesn't matter as long as the grounds not frozen and could be washed away by rain.

There is no such thing as a Empire Zoysia fertlizer. The sod farm website gave you generic information like I do for any grass using a 4-1-2 or 3-1-2 fertlizer for the first spring application in lieu of a soil test. 4-1-2 and 3-1-2 are what is called a natural or natures balance of the 3 main nutrients. To know what you really need, only a soil test can tell you what is really needed. Your soil test could come back and tell you are extremely high in Phosphorous and Potassium. If that is the case you will not need any for a few years and only need straight Nitrogen. As a generic recommendation you use a 3-1-2 ratio for the first application in spring, and then just straight nitrogen for the rest of the applications that year. So for Zoysia one app of a balanced in spring, followed by one application of nitrogen about 10 weeks later and you are done for the year.

Perhaps too late to help BUT 1. It looks like the neighbor does not have the same problem. Since, the poster says she was not home, I would ask them what they are doing differently. First of all you can learn what works. Perhaps, more important, you politely tell them you are trying to correct the problem. 2. Does not seem to be a too much water issue as from the photo that low spot behind the mail box would be more wet than the higher parts in the picture. 3. Not sure how large the lawn is but, for dandelions the commonly sold dandelion digger (fork) if properly used-cut root well below ground gives instant permanent chemical free results. 4. Dandelions are easy to kill with the common broad leaf weed killers. You need to be sure the instructions say OK for zoyzia. The problem is these products take about two weeks to kill dandelions in that much time those yellow flowers will turn to the grey puff ball seeds so by the time the chemical works the dandelions will not only seed your lawn but neighbors for miles around. 5. DO NOT used the commonly sold weed and feed products on zoyzia all that I've see are far too high nitrogen for empire zoyzia. The liquid concentrates work faster than the powders and are less costly to use. I spot treat with diluted spray in a windex type trigger bottle. It seems to last forever. I think the qt I have is at least 2/3 full. It still works fine and I bought it at least 5 years ago.

Please reconsider St Augustine. Unless you are growing prairie grasses like buffalo, blue grama, or the wheatgrasses, all the normal turf grasses require roughly the same maintenance (weekly water and mowing with 3x per year fertilizer). Zoysia might be different if you lived in Korea, the Philippines, or Hawaii, but not in the main US. If you don't care whether it looks nice, then you can let it go unwatered without killing it. It will go dormant for the rest of the growing season and come back from dormancy next spring. All during the dormancy it will be subject to uncontrolled weed pressure. The fact that it returns to life after going dormant is what the zoysia proponents refer to as "drought resistant." St Augustine is not drought resistant because it has no drought dormancy state. If you allow St Aug to go bone dry for 3 or 4 weeks, it will die completely and not come back. So, as I said, if you want the grass to look nice, you have to water it. The best organic fertilizer for YOU is whatever ground up nut, bean, or seed you can find cheap at your local feed store. For me that is alfalfa pellets and corn meal. In South Texas both of those are under $10 this year. For people in the north it is soybean meal. Milorganite is usually available everywhere. I don't use it but other people I trust swear by it. The application rate is 15 to 20 pounds per 1,000 square feet. You can apply any time you want. I try to follow the federal holidays. I start on Washington's Birthday, then Memorial Day, 4th of July, Labor Day, and Thanksgiving. I can remember those dates and they keep the lawn green. Never till in preparation for grass. Just don't even think about it. Level the area as much as you want at the surface by dragging a piece of chain link fence or a 2x4 around. It might need some weight to actually grab the soil and move it, but that's all you need. Plant the sod right on top of the leveled surface. Then roll the sod down to ensure the bottom of the sod (the roots) are in contact with the underlying soil surface. Roots will not grow through the air to reach the soil.