Carly -
My only concern, for the doggy, is his air travel. You will probably travel with him inside a kennel (in the baggage compartment) which is a dark, extremely cold and noisy place in the airplane in flight. I would suggest you put him in his kennel with his favorite dog toy, and maybe his winter coat if any. If you have to change airplane, he might be cooking in the sun while waiting to be loaded on another plane.
xxx
If you love your doggy, get him to travel with a dedicated cargo airplane, such as FedEx or UPS plane. In these airplanes, live domestic animals are loaded on the forward main cabin pallet, where there is light, air conditioning, decent temperature and less noise. It is likely that one of the pilots will sneak out of the cockpit maybe to smoke a ciggy (illegally) and when he sees your little doggy, will give him a pat on the back, or a cup of milk and a piece of a crew sandwich.
xxx
I am a retired airline pilot. I occasionally flew cargo airplanes, we had animals at times and this is my personal experience. I am an animal lover, and would send my doggy or cat as I suggest. Just get his travel to match your own schedule and put written instructions with the airbill about the dog with a contact phone number for arrival, and if need be join some instructions in English and Spanish for the arrival in Ezeiza. See a vet also for a dog tranquilizer... if you think he might need it.
xxx
In Buenos Aires, go to Carrefour market or Walmart for dogfood - Unlike USA, dogs are welcome in many restaurants, just ask a little dish with food or a small treat to be served for him under the table... I have a husky, he loves beef empañadas and I order one for him sometimes. Go to the parks with your dog, like in Palermo. Many students walk dogs to make a little money, and have half a dozen of them for part of the day to these parks.
xxx
Taxis will take dogs, at worst driver might ask you to leave him sit on the floor. Better than a taxi - look in phone book for a "Remis" (re-meess) nearby, they are unmarked private taxis (request a quote for your trip = often almost equal to regular taxis - and if need be, arrange for a return trip, time and location) - For "I have a doggy", just say "tengo un perro" in Spanish if you want to say it over the phone.
xxx
Enjoy your stay...