That's really an ambitious amount of miles and places for the amount of time you have.
You'll need more than 1 or two days for Yellowstone.
I've found that our average speed for a days travel will be about 50 mph with fuel stops, lunch and breaks.
Try setting your plan to paper figuring how many days you want to spend in each place as you go and including travel days. In doing that you will begin to get an idea how much you can really do.
If you try to do too much you'll all be too tired to enjoy it.
further in this forum is a section "Travelling in Canada and Alaska" in there you will find links for requirement for travelling with the dog. You won't need a health certificate, that's for commercial purposes, you will need proof of rabies and some other vaccines and current license.
I hope you can do this and have a safe and enjoyable trip.
Milt