Turists can't get a CUIT, it's a CDI.
Plenty of threads deal with the subject:
1. Go to the Police station in BA (or Registro in the Provincia) and ask for a Certificado de Domicilio. Cost is like 10 pesos
2. Go to the AFIP branch linked to your address (and only that one), and ask for a CDI. That's free
Then you can buy a car the usual way (Registro automotor, etc.)
Exiting the country with your own car as a tourist = used to be impossible but it seems it's possible now.
Buying a car in Chile: why not but you'll need to take the car out/in of Argentina every 3 months unless it'll get seized.
Added: no need to prove the money origin. It's indeed interesting to lower a bit the price of the vehicle purchased in the declaration but everyone does it so the price must remain rational (they are no fools).