You can have a great time down here My responses...

No suggestions, never used one of the language schools
Recoleta is the most tourist-friendly neighborhood, but you can find better deals in the other neighborhoods. San Telmo, Palermo, and Belgrano are all good ones.
Iguazu is the first choice (see my recent post), especially if you come in April, the water may be a bit lower in November due to seasonal rainfall. Mendoza, Jujuy, Corrientes, Rosario, Ushuaia, Salta and other cities can all be interesting, if you can explain what you're looking for we can make more specific recommendations.