Unfortunately, a lot of socializing involves going to restaurants here, which are great but by no means super cheap, even by US standards.
For 1. Always take the bus/subte/bike to your destination activity.
I second the Costanera; this weekend the SO and I took a calming walk around the Nature Preserve. They had all kinds of music around the Costanera this weekend. There were even salsa instructors giving free lessons around the Avenida Independencia entrance to the Costanera. We brought our own mate and walked around the trails of the nature preserve, nice!
Also, check out this website
http://www.buenosaires.gov.ar/areas/cultura/
It is the City Government's culture website, which will keep you up to date on all the latest activities, most of which are LIBRE Y GRATUITO. Every neighborhood also has a Centro Cultural. This city is packed of creative, talented, brilliant porteños, a lot of whom offer shows, art presentations, and theater for next to nothing in cultural centers. I would go down to your local Cultural Center and ask if they have a calendar of upcoming events, most of which will be FREE! Go on a bike outing for beginners, leaving every Saturday meeting at Godoy Cruz y Av. Libertador at 14:30 for a 35 kilometer ride there and back. I like bike around the city and explore and that is FREE if you can remember to pack your own food and drink and know where your bathroom stops are. You can register and check out a FREE BIKE to use in the city at one of the bike stops.
http://mejorenbici.gob.ar/?id=5 You can even use a passport to check one out. I am not sure how this rental process works but worth checking out.
Look around you...Argentines are great at having a great time on the cheap, sitting on a building stoop, chupando with friends (I guess technically illegal to drink a Quilmes on the sidewalk) or having parties at their friends' houses! Take inspiration from them. I grew up in a household where I "consumed" my entertainment; I learned you had to pay to play. But fun can be free when you get creative!