When you are watering the lawn and or it is raining such as in the spring/summer, the nitrogen and salts from the dog's urine is getting diluted which may be why the problem is much less apparent during those times of the year. Gypsum, baking soda, lime, etc are salts and can actually compound the problem. As you mentioned, getting the dog to go somewhere specific is the best way to handle this problem. It can take quite a bit of training to re-teach the dog where to go. As someone who has had to re-teach rescues about potty habits, it can take some time and work. It might be helpful to consult with a dog trainer - the Humane Society has trainers and experience with rescues, so that might be a place to start. On to the grass that is already damaged - really irrigating those spots and washing out the nitrogen and salts is important. By doing this if the turf is still alive, it may recover. It also will better prepare the area for re-seeding. If the turf is totally dead, removed the turf and 1/2 to 1" of soil and then re-sod or re-seed. In a Kentucky bluegrass lawn: Spot seed with Kentucky bluegrass (marginally effective) or perennial ryegrass (more effective). http://polk.uwex.edu/horticulture/seasonal-concepts/lawns-and-dogs-but-not-tomato-juice/