My partner and I brought Mister the Westie down with us 3 years ago, and he's travelled back and forth between Argentina, Europe, and the US frequently. The rules vary a bit depending on where you're going. I can add data for the US if anyone needs it.
For Europe, the animal should have a microchip following EU standard. (Mister now has 2 - one for the US and one for Europe.) You'll need to take the animal to SENASA, by the Puerto, so they can verify the chip and complete the EU paperwork, based on the certificate your local Vet issues you.
Once in Europe, you may be asked to produce the paperwork on entry. We've arrived with Mister variously in Frankfurt, Amsterdam, Madrid, and Paris, and I've only once been approached by anyone interested in the dog. That time an agent in Frankfurt asked me if I had the paperwork. I replied "Yes," and started to dig it out of my bag, but he waved me on.
The airlines tend to be more demanding of this documentation than the authorities. Before boarding an animal, they usually inspect all the paperwork in great detail. At Ezeiza, they make copies of it and charge an inspection fee.
The EU now has a Pet Passport system that you can obtain at any registered Vet in the EU. It simplifies things a bit, although I'll admit that no one has ever asked to see Mister's passport (which is in Dutch!) I have found some airlines in the EU will not board a pet that does not have the standard EU passport.
As for the elaborate Rabies tests, these are required only for the UK and a couple other countries we never visit. The required documentation is virtually impossible to obtain and then to maintain if you don't live in Europe or visit at least once a year. If you're going to continental Europe, you don't need it, so don't worry. (This was England's revenge on the EU's requirement to allow visiting pets, which had never been permitted in the UK without a long period of quarantine.)
Finally, the prohibition on 3-year Rabies tests is an Argentine law having nothing to do with regs outside Argentina. It's illegal to have a pet here that does not receive annual rabies shots. Although most other countries require innoculations only every 2 or 3 years, Argentina maintains that there is sufficient incidence of rabies here to warrant annual vaccinations.
Should your pet claw or bite anyone in Argentina, you'll be required to produce a vaccination certificate no less than 1 year old and to submit daily observation reports from your vet to the Insitute Pasteur, in the Parque Centenario, for 10 days following the incident. Failure to comply requires destruction of the pet.