NSTs do not have a uniform set of rules. That said for the major trails, there are good websites that can answer many of your questions.
The ATC website says that dogs are NOT allowed in 3 areas: Baxter State Park in Maine, Great Smoky mountains National Park in Tennessee, and Bear Mountain State Park in New York. There is an alternate road walk for the Bear Mountain section. The BSP section can be done in a single day and it is the first or last day which can make logistics easier. That said, the remoteness of BSP makes logistics more difficult. For GSMNP you will need to find someone to take care of your dog for the 4 or so days it takes to hike through the park.
The PCTA provides information on the PCT. While I cannot find anything that says dogs are allowed, they are (along with horses). That said, hiking the PCT with a dog is a bad idea because of the long distances between resupplies, water shortages, weather, terrain, bears, and snakes.