Interpersonal skills, lab dynamics, and personality
It is incredibly unfortunate that you've found yourself in a situation where it doesn't sound like there's much you could have done to actively avoid the contention. I do feel like there must be something in the story that you're leaving out, but if I base my conclusions assuming you've included everything that's relevant, here's my two cents worth of feedback to your questions:


Being the black sheep does not mean that you should leave the lab. In fact, what you are experiencing is not unique and is something that generally happens to people throughout their lives (to more or lesser of degree). Simply leaving the group might be an easy and acceptable solution to your current situation (only you can really judge that I think), but you might want to view this as an opportunity to build your conflict resolution skills, which will be incredibly valuable throughout your life.
This is not going to ruin your professional life, but like with all events in our lives (whether fair or unfair), it will absolutely alter your career-path experience. In general, it is always best to have a pro-active mindset (e.g., "I can direct my own career path") rather than a passive mindset (e.g., "these people will determine my career path"), regardless of whether other people can influence your trajectory.
Whether or not this is your fault, you certainly can learn skills from the experience. In my opinion, the best thing you can do right now is to focus solely on how you can serve your colleagues. Having a dialogue with them could be useful, but like you said you'll mostly be at their mercy in that approach. If you proactively look for opportunities to help them in their own work and learn the ropes in the lab on your own so you can help them and generally be a "good neighbor", then they will really have no reason to maintain their animosity towards you.


I'm not sure there's much you can control beyond my points above. An enormous skill to learn in life is to not stress out over things that you can't control. Another enormously valuable skill is being able to recognize what you can or cannot control around you. Proactively seek to make the things around your more amenable to unity and cooperation, regardless of others' reaction to you. Be sincere in your efforts, not superficial or only acting to get a specific response.