There is a guy I've seen in Austin a couple times who rides with a poodle that stands on a small platform on his top tube. He was actually at a large group ride I did on January 1st this year and I can vouch that it's for real.
Here is an article in the paper:
http://www.austin360.com/recreation/...rm-742332.html
As far as finding places to stay, lots of motels are now pet friendly and there are online directories with that info. Campgrounds usually allow dogs if you keep it on a leash.
I have 2 dogs that I love to death. The biggest problem with them is when we want to travel, so considering that you are wanting to travel, getting a dog might be more of a headache than you want. But, you can make it work if it's something you really want to do. Or, if you're looking at doing a cross country tour as a once in a lifetime type thing, you can get a dog when you return. If you decide along the way to tour for a lot longer or you want to do more longer trips, then you're not stuck with a commitment to the dog.