You can look on
craigslist to get a rough idea of what it will cost for a tourist style temp apartment. These tend to be very expensive and in good areas, so don't take it as the only source for info or prices.
Is it a good idea to pay through the nose to have a nice place and not work hard to get it? Sure, if you've got money, which is true anywhere in the world. I'm not sure what your choices are anyway since there's 3 of you and your comfort level is an issue. Personally, I'd just stay at a hostel but that's just me.
There's literally thousands of
blogs about Buenos Aires alone, not to mention a few thousand more about Argentina. Those can be good sources for info. Also, look on the mainpage here at baexpats for a list of articles and their source websites or blogs. This subject has also be a hot topic many times in the forums, even last few weeks or so I remember a heated debate taking place.... i.e. do a search.
The pay for teaching english? This depends if you're fulltime or temp, if you have a work visa and DNI, if you have qualifications and lastly, if you're just walking in off the street and looking for adventure. An australian guy I met in BA got paid 1200 pesos a month, which translates to $400 U.S. dollars PER MONTH. Is that a lot of money to you?
That would be the low end of course. Other people who teach to business people or run their own schools can make more.
As I said before... there's a glut of teachers and for some reason every new arrival has this dream in their head. I don't know what your motivation is but if it's to make your fortune or even survive in BA, then teaching is about the worst choice you can make. Spending money on a TEFL course is another bad choice but then again, maybe this is a career you dream of doing elsewhere and you see something bigger happening. There are other countries where teaching english does pay quite well although even that can be rough. Japan just shut down a ton of schools yesterday and Korea cracks down every few years.
Good luck and keep doing research.