I would expect problems in Kentucky and Missouri. We were chased by dogs multiple times a day in both those states. I would have to guess we were chased by at least 30-40 dogs and probably more, sometimes 2-4 of them at a time.
I know that day after day we had 100F heat on our TA. Most dogs do not do well running in 100F heat. I have a dog that runs with me and I wouldn't make very good time trying to cross the country at her pace and daily mileage in hot weather.
Personally I wouldn't expect my dog to go faster or farther than I would on foot. A lot does depend on the breed and age of your dog though. If I were going to take my dog I would either let her ride in a trailer on the flats and downhills and get up the climbs under her own power or I would skip the bike and walk/run carrying my stuff in a baby jogger.
As far as how often there are vets offices... There won't even be towns closer that 40 miles apart fairly often and once I think they were 80 miles apart. Some of these towns had a population of less than 50 people. You may well be several days to a week's ride from a vet at the pace I would be willing to push my dog. And if she was sick or injured it would be ugly.
You will have difficulties in some places where dogs will not be allowed. I am pretty sure you won't be able to take fido into any national parks and maybe some state parks. You can probably work around this one, but it may be a hassle at times.