I took my Shepherd, Nesta, up Noonmark a couple of times. It was one of his favorite mountains. I checked the ADK HP guidebook before bringing him the first time, for I knew he was not allowed to gallop with me down the main road, and in my book (on page 75), it clearly states the boundaries of the AMR, and the approaches that are exempt from the AMR restrictions: Snow, Hopkins, Giant, Noonmark, Round, Dix, and LWJ (if approached via Deer Brook)
You can't bring dogs on AMR property, which starts at the wooden gate on the road. The one with the "AMR" on it.
As others have stated, and the vast majority of us do, please respect and adhere to those rules and restrictions.
However, the Noonmark, Round, and Dix trailhead (Stimson Trail) are on private property--just not AMR property. Check your maps and you'll notice the line of delineation between the two private properties. So, you can (or could in the 90's when I went with my dog) take your dog for a loop over the two. Of course all the usual dog etiquette and trail rules still apply, and are greatly appreciated by all of us who hike. There is nothing worse for dog lovers than seeing disrespectful dog owners out there.
As for the mythical trail damage of dogs over humans ... it's all physics. Nesta was 70 pounds spread out over four spongy padded paws. I was (am) 200+ pounds directed and focused onto two feet clad with Vibram hard rubber soles. 6X the weight impact and how many times the footprint of hard rubber versus the softest part on a dog's body? 10X? 25X? 100X?
Just do all of us dog lovers proud and keep him leashed, under control and don't leave any "doggie presents" behind.
Happy hiking to both of you!