Assume that my Argentinian car rental company allows me to drive into Brazil: Is it possible for me to cross the river and drive from Puerto Iguazú to Foz do Iguaçu to see the Brazilian sides of the waterfalls? Are there any common pitfalls?
As long as you have a valid passport and you are allowed to enter into Brazil from Argentina, yes, you can do that.
When I went to the Iguazu waterfalls I was staying in a Hotel in Brazil, but I spent a couple of days on the Brazilian side and a couple of days on the Argentinian side. I passed the border a couple of times per day without any issue.
Just keep in mind that, depending on the season, the queue to cross the border can be quite long. You want to plan between 15 to 30 minutes just for that (or more).
Also, make sure to plan in advance about your car parking. I was with a guide and I did not have to worry about that, but especially for the Brazilian side I'm not sure you are allowed to enter with a private car inside the Park. You may need to park it outside.