In Boca Chica you do. And you get to wear this big ugly carnet to prove it. And Politur will run you off if not. I have no idea if non-residents can get one - I doubt it and you'd be violating a tourist card declaration as well if you said the visit was for tourism.
Not "perhaps not". Look at it this way, if a Dominican , after the 1/100,000 odds of getting a tourist visa to the US started working in the US he'd be deported. At least in this case you'd probably be told to stop. Unless Politur was feeling particularly nasty and arrested you.
