The dogs and the lawn

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Firstly never mix newly seeded lawn with dogs. In fact, on the early growing stage of the lawn NO TRAFFIC is recommended. 
 
Secondly, only omnivores animals which eat plants produce excess  waste with nutrients that can be assimilated by plants. 
 
No carnivorous can produce useable waste for plants.  
 
Anyway, before grasses can efficiently handle pet waste and trails, it must be allowed to develop strong root systems and knit tightly. The same goes for the sod too. 
Urine spotting
As one of the most  frequent problems of lawns that have a pet, it has a greener ring than the lawn and the center is yellowish. Thats because the amount of ammonia from the pet's urine is smaller on the edge of the spot, acting as a "fertilizer" and making the grass greener, as in the center being too high and killing the grass. 
Urine spots can be found all over the lawn, as dogs don't prefer particular spots. 
Train your dog to use another area beside the lawn (best when puppy) or simply don't allow your pet free rides on the lawn. 
You can also give your dog some dietary supplements that will bind the ammonia from the urine, making it harmless for the lawn. They are made with natural plant extracts, have no chemicals and do not modify urine pH, thus not altering kidney function. 
 
Lawn wear 
As they are looking for attention, dogs run from one part of the yard to another on a well establish pattern, seldom straying off it. Their regular walk will produce wear patterns, causing the lawn to thin over time, and even die. 
The best you can do about this is to forbid your dog to walk freely on the lawn or replace grass from the trail with mulch, bark, gravel or even pave it. 
 
Hole digging 
This dog habit that occurs at young ages and with playful dogs is the most dangerous of all for your lawn as it will destroy it completely.  
The only solution is to train and discipline this habit out of your dog's behavior.
 
Dog-tough grasses
As a rule of thumb, grasses that spread by stolons or rhizomes tend to regenerate faster and are more fit for re-seeding damaged areas in the lawn. 
 
Warm season grasses 
Best choice of all is bermudagrass as it handles the traffic efficiently and it regenerates fast. 
Ryegrasses are a good choice for overseeding on bermuda grass as they provide additional protection and color over winter. 
Perennial rye-grasses are also a very good choice for a fast cosmetic repair.
Zoysia handles traffic good but it takes pretty long to repair. It takes to establish almost four years in the first place. 
 
Cool season grasses 
Due to its ability to repair and spread over winter, Kentucky Bluegrass is the best choice for northern climates.