If your dog has an international health certificate showing recent shots, you will be OK entering Mexico; however, other countries may have different requirements and you should investigate each one for what they are for you, your vehicle, your possessions and your pet.
The RV idea will help with the 'dog problem' but you should know that campgrounds are not what you might expect in the US or Europe. Often, you will be staying at a Pemex gas station, in front of the local police station or wherever you can ask and gain permission. Some hotels & motels have parking lots with electric hook-up and will allow you to use a shower. Look for a camping guide to Mexico online but don't expect it to be up to date. You will seldom have electricity sufficient to operate an RV air conditioner and that can be onerous.