You might consider living in a neighborhood like West Asheville/Haywood Road or North Asheville/Merrimon. There are parks to walk the dogs in walking distance and probably less of a noise issue than downtown with sirens. There is not the issue of aviation noise in downtown or surrounding neighborhoods to consider. As far as pet owners, occasionally there will be irresponsible owners on trails or in dog parks as the other post said, but for the most part, people and pets are pretty reasonable and friendly. As far as trains, stay away from Biltmore area. Good luck & hope you find what you are looking for!
It depends where you live....There are many people who do not take responsibility for their dogs as far as I can tell, but I don't hear a lot barking where I live. Lots of people letting their bully-ish, aggressive dogs off leash though on trails, so be careful of that if you have a dog. Lots of sirens is the most annoying thing and a lot of congestion traffic wise for a small city so lots of motor noise - again depending on where you live...I am sure there are quieter areas, but I have heard people complain about sirens. I never hear air traffic noise.