Hi Emily,
As far as dog-less riding goes, I live in the far NW suburbs of Boston, a little west of Concord, MA (you know, the Revolutionary War, etc). All of the small towns around here have leash laws and I have had very little trouble with dogs. The only major thing I had was on a ride with my husband and another couple, but it was in far central MA, almost western MA, about 45 miles west of here, out in the real country. Around here I can go out my door and be riding in low traffic areas, with lots of hills. All of the clubs around Boston have their weekend rides out here; there are so many cues on the roads that everyone notices them! Of course, only hardy souls ride in the winter, but there are tons of recreational/sports things to do besides cycling.
I also just started moutain biking and I am surprised that you see it as safer! Yes, there probably won't be dogs on the trails and the fat tires do give confidence, but around here mountain biking means skinny trails, rocks, and roots. There are some fire roads and flat paths, but I find it very challenging compared to road biking.
Good luck!
