No, the monsters ancestors were mistreated by cavemen. When they were transformed, they started scaring humanity in an act of revenge and became all the monsters in mythology, though they had a longer way to travel before the door system was created. That "Sulley carving" in the Witch's cottage is probably a memento from when the witch (who is really no different from Merlin or Genie as far as "Disney Magic Users that spout anachronisms") made a deal with an ancestral Sullivan and taught him how to create magic doors, allowing monsters to go between the human world and the monster world pocket dimension easily. "Supernatural" beings have to look out for one another after all with humans starting to expand.
There's nothing in any Pixar movie to imply time-travel, that guy was pulling a bunch of stuff out of his butt. Also, any of those theories that force Cars in as a post-apocalypse thing are just forcing a square peg into a round hole. It's just an alternate universe or maybe what Hot Wheels experience during playtime ala Toy Story 3's opening.