I don't go to Columbia, was just admitted yesterday, but I have lived in NYC for the past 2.5 years and I can tell you it is very expensive, and I would not recommend commuting from Brooklyn to campus. The closer areas in Brooklyn do not differ significantly in rent from NYC rent once you get so far North anyway. In terms of students being cutthroat and self-centered, I attended an ivy undergrad and did not experience anything like that and I am sure that the environment is totally different once you get into grad school.
Definitely second the advice that you shouldn't live in Brooklyn; currently live in the Northern most part (where I grew up) and it takes almost two hours to get up to Mailman's campus, given it's a slightly awkward neighborhood to reach by subway. You can find decently priced places, even comparable place in the Heights/Inwood area, or I would say you are better off going to the South Bronx if you want to save money. I haven't heard anything about Columbia subsidizing metrocards, but I doubt it. Buy a monthly unlimited for $104 and it'll bring you much farther than that amount would in gas.
I've spoken to a lot of students at the school and everyone has been incredible nice, informative and too busy doing their own amazing work and projects to be mean to one another. That being said, a lot of students and faculty do their own thing very often- you have a lot of people work full time or part time and taking classes, a lot of professors working with other agencies and traveling- so you have opportunity to do a lot of cool projects and research but I've heard some people say they wish there was a stronger campus community.I know a lot of people take classes in other schools, especially SIPA and Social Work, which can give you an opportunity to explore Morningside Heights, but Mailman itself is pretty confined to the Heights (all classes, professors, and research happens there).
As for cost of living, there's no denying it - NYC is one of the most expensive cities in the country. This is rent, entertainment, food, etc. This is not to say you can't live on a budget. Don't eat out too much, use farmer's markets, Trader Joe's, Fairways, BJs and the like. There's a lot of free events and deals and gems all over this city if you're willing to look - in the end totally worth your money. Hope this helps and good luck deciding!