First of all the Gemora already mentions the issue of fear of dogs by stating that one may not raise a vicious dog in the city as the fear of it may cause a pregnant woman to miscarry. However, I gather that your dog is not "vicious". Furthermore, a dog is a chaya temei'a and in Kabbala the dog often represents the forces of tuma. However, I think that the visceral reaction you experience is based on hundreds if not thousand of years of anti-Semitic Goyim using dogs to attack Jews. While you may feel that the reaction is irrational and excessive, understand that it is sincere and please make every reasonable effort to prevent your neighbors' distress.
I'm not familiar with any Halacha regarding petting unfamiliar dogs, but common sense dictates using caution around any stray animal.