Ok, I have a better picture. It seems that you want to explore and work while you're doing it. Great approach, especially being able to work online. Most cities of any size will have cybercafes and some will have coffee shops with WiFi, so that will be relatively easy. When you rent for several months, you would have to get a phone in your own name in order to have DSL from Telmex. If you were in an area served by Cable, that would probably be easier and also faster.
You will still have a problem traveling with a dog in Mexico, but you'll have to learn how to handle that as you go. In cases where you just want to sleep (you can't leave and return to the room) you can use 'no-tell' motels. They offer total privacy and security for up to 12 hours and are cheap. Just walk the dog before checking in at bedtime. Since you never come face to face with management, you can take the dog in discreetly.
If your Cherokee has the spare on an external bracket, have it welded to the door and add locking lug nuts to it and all your wheels. A club, alarm and a kill switch are recommended.
When you are attracted to any particular place to settle, be sure to visit it in all seasons; some are intolerably hot & humid for more than half the year. That makes the coastal areas undesirable for most folks. It is better to look into the central highlands of Mexico for climate. I can't advise about Central or South America, just Mexico and, particularly, the Lake Chapala/Guadalajara area, where we have lived since 2001. We are within half a day's drive of Pacific beaches and only visit them just before, or just after winter high season in order to get affordable prices. You can find some villages on the beaches where you may be able to 'camp' economically but many are not recommended.
If your online work is billed from the USA, paid there and deposited there, your only tax liability will be in the USA, as usual. Nothing will change in that regard.
When you enter Mexico, you will have to temporarily import your car. Be absolutely sure that you reverse the process and keep all receipts for the sticker whenever you leave Mexico.