A mental health assessment can be as short as an hour - several hours if formal testing is done. Therapists can go through a list of symptoms and come up with a working diagnosis in about that much time - first or second visit. Sometimes personality disorders are not evident at first and could take months or years before they are recognized, especially if the therapist is not specifically looking for personality disorders. Quite often they aren't, since primary complaints are more likely to be depression or anxiety or something with more obvious and immediate effects, and because insurance usually doesn't cover personality disorders. Also, a lot of therapists are reluctant to diagnose personality disorders because they can be negative labels with a lot of stigma.

Definitely, you should tell them about your concerns and realizations. It is your treatment, and your responsibility to choose what you feel is most important for you to work on. The therapist is there to guide you. You could choose to work on understanding your behavior patterns, changing any that still continue to harm you or people you care about, repairing relationships and finding new relationships - also accepting yourself as you are now and making sense of your life.

They may or may not agree with you about having BPD. If they disagree, it doesn't necessarily mean that you are wrong - mental health diagnosis is still pretty subjective. They can still be very helpful with your concerns. Longstanding or recurrent depression comes from somewhere - your past experiences, genetic predispositions, social environment, how you think about yourself and your life, biochemical stuff, etc. That is true regardless of what diagnosis they assign to you, and those are the kinds of things that will be addressed in treatment. Getting a diagnosis is only a start towards deciding what kind of treatment is most likely to help you.

The idea of group therapy is scary if you have had problems in social situations and with relationships, but actually is a really good way to address those problems. The other people in the group would be people who have similar problems and fears. In a group, with support and guidance, you get the chance to experiment with new ways to relate to other people, and learn what works better for you than what you have always done. You might not like everyone in the group, but you probably will like most of them and they will like you too. If conflicts happen, that's what the group is there for - to learn to work through them so that you can also work through similar stuff with the people in your life.