thefishwife my girls are actually pretty good, but I am the same -I'm not prepared to do extras like ferry around, give money for clothes, going out and so forth for people who are downright rude or dismissive. I'm happy to stop contributing if I'm treated like shit and so far, this line in the sand has held. I don't fuss over a bit of sighing, the odd door slam etc, but things like being directly rude to my face aren't ok, and I expect an apology before moving on to have it not happen again (too soon!)
You do have to move on quickly though and not bear a grudge or take it personally, their huffing and puffing and criticism is often then directed at themselves (I'm so fat/ugly/not good at school) as much as at others.
I try to do one on one things, sometimes it works, sometimes they don't look like they are enjoying it much! Watching the same TV show curled up on the sofa is a good one, and one reason I don't like everyone being on separate screens. We usually watch a few shows/films on the weekend together and talk about them- usually their choices. Ditto being in the car, going out for lunch.
If everyone is rude and not helping, then I might sit them down and re-set the boundaries as I can see three is a gang and they may be tag teaming being little shits. If so, remember you have the money, the internet password, the car and the will to help them and make their lives fun and interesting, so the least you need in return is reasonable politeness most of the time, excusing the odd lapse.
I don't really agree with some of the teen brain stuff as it's too dramatic, some teens are off the rails, but clearly they can behave in school and to relatives, why should they get to take their nastiness/shit out on you? I cringe when I hear how some of friends let their children talk to them and don't even attempt to challenge it, but just shrug their shoulders when called a bitch or whatever.