I think your daughter needs a therapist, and you possibly need a different therapist. Wanting attention from people on instagram so badly that you cut yourself (even shallowly) is a giant issue, and you also reference underlying depression. Also, you're vibrating all over the place with concerns about what your daughter's mental state means about you as a parent - what did you do wrong, how did you mess up. That's not helpful to your daughter in processing her issues. She needs to talk to someone who sees her issues as entirely about her, without an investment in how they reflect.
What was the point of having your daughter miss an exam? Surely she was not at risk of cutting herself *while* taking a final. She could spill her guts to you just as well after passing her class.
"Hours of questioning" tends not to get you the truth so much as what the subject of the questioning thinks you will find narratively satisfying.
I also - HUGELY - think you should stay out of it about her boyfriend.
I question your conclusion that the boy who pressured your daughter to send scantily clad selfies now now now is a "good kid." He can act good sometimes, but right there? That was a bad thing for him to do, and a potential relationship dealbreaker. He treated her badly. Acknowledge that in your head. And when you take her to the mall to walk around with her boyfriend, exercise some reasonable skepticism about his feelings and intentions. Feel free, for example, to be less than thrilled with facilitating the hangout. If you want your daughter to respect herself, you have to at least raise your eyebrows when people treat her badly.
However, parental interference in romance (teen or otherwise) tends to just build the drama and cement disasters. Your relationship is with your daughter, and it is not your job to advocate for her boyfriends. If she wants to break up with him, she should. If she wants to feel out whether her friends would support her or still like her if she wasn't dating this guy, that's reasonable research for her to undertake. If she's insecure about whether her boyfriend's interest will continue now that she is prevented from sending pictures of herself in her underwear - well, if that's a dealbreaker for him, your attitude as a parent should be 'good riddance.' (Incidentally, you do not help her feel secure in her relationship with her boyfriend, especially not this boyfriend - she's not secure because - possibly among other reasons - he's not all that safe or pleasant to date.)
Teenage romances are often spectacular emotional dramas, and it's exhausting, but I am going to argue that this is age appropriate. Better that kids should understand what it's like to be too intense and overreact and fight and win and lose and communicate badly as teens, really. The stakes are low. Communicating badly and having a stupid breakup in high school will hopefully help her understand how to communicate well and keep a relationship going when she's ready to do that, which she isn't (because she's fourteen). Having her own personal flameout will be far more educational than having you lecture her and attempt to dictate how she deals with her boyfriend. Flaming out and getting back up and going to school again will also be an important lesson to her, when that happens, about how relationships sometimes end, but life goes on.