Elk - Can't provide advice about the bike, however, I would strongly discourage you from riding with your dog attached by a leash to your bike. I've seen people try to do this and it's not safe, particularly on a shared trail. Most importantly, I always feel sorry for the dog, who struggles to keep up with its owner. It's dangerous, also, for anyone else who is either riding their bikes or are on foot. You can never predict if a dog will take off after something, pulling you down, run into someone else (causing another cyclist to wipe out, for example), etc.

Conversely, a guy at work mountain bikes in the woods and his dog comes w/him. The dog is not leashed, however, so the dog is able to take breaks and go at his own pace.