I also started going to the mountains just because of the dog, now it's totally hooked me and I really enjoy the mountains but I can't imagine going anywhere without my faithful companion, so of course I won't go where you have to climb a bit, but there are enough other hills where we can both enjoy without fear.
Since I'm a student I have the chance to go to the mountains also during the week when there aren't so many people, in any case I prefer paths with less traffic. Even popular peaks usually have paths where you don't meet a soul.
I myself haven't had bad experiences because of the dog yet. At the top I always leash her, if we meet someone on the way I move her off the path and she sits nicely while people pass. Sometimes people are right around the corner and I can't move her fast enough but she doesn't push into people, in such situations she goes by nicely.
Signs though.... no comment. Not only have I noticed "dog on leash" signs, on the path from Ljubelj to Zelenica even that the dog must be on leash and muzzled (mid summer, in the sun, hiking uphill and don't allow the dog to cool with the only possible way - tongue), on one path to Gonte I even noticed a sign (middle of asphalted road) "forbidden for dogs and cyclists". People are impatient, I don't want to be a conflict person and I know I won't get anywhere arguing on the forum, so I ignore such posts.