Well, I think the idea of not giving a fuck isn't to literally not care what anyone thinks about you, period. Going about it that way has its own set of problems. I think the more important thing is in how you deal with peoples' opinions of you. First, I think that you need to have an opinion of yourself, and I think that opinion needs to be generally positive. If it's not positive, focus on the things you don't like and how to fix them. But the important thing is that this opinion should be cultivated from your own feelings and conclusions. Once you have a foundation of self-esteem that you've given yourself, I think it's easier to react to what other people think and evaluate their reasoning against your own. Without that, you'd just be riding entirely on what you hear from others as a means of self-evaluation.

I'm not very good at talking to people I don't know, so I'm not sure how much my advice is worth in that regard. But I think, in general, asking people about thmselves is good. There are certainly questions that are too personal for introductory conversation, but the general things like what they do and what their interests are are fine. People, generally, like to talk about themselves. So I think you often won't come across as nosy just for asking questions, so long as your questions are appropriate. As for approaching people, that's my absolute worst subject. But I think proximity helps. I think you try to read the situation that person is in, and if there's an opening to say hi to them, you go for it.

Damaged relationships generally benefit from truth, one way or another. Especially in the case where you don't know what's wrong. I would recommend just asking people if if there's something wrong between the two of you, or something they don't feel comfortable about in talking to you. It might be uncomfortable for them to talk about it, but I think unless you know what's up, or if anything even is up, it's hard to know what to do.