We saw a lot of what you describe with your dd in our oldest (now 21). She was a fussy baby who grew into a precocious, delightful child at an early age but grew more and more intense as she got older. I don't remember the exact sequence of things, but I do remember how hard she took news that we were moving when she was 8 years old/4th grade. The following New Years Eve, at 9 years old, when she missed the "ball dropping" due to a misunderstanding (long story, we were watching a movie, had asked the kids whether they wanted to switch back) she had a two-hour meltdown (yes, til 2am) and I can remember thinking, "this isn't just normal kid stuff." We had definitely encountered things before, but I remember that being the straw that sent us over the edge and into counseling. We have had a few diagnoses over the ensuing years: depression, anxiety (GAD), SAD, and even bipolar II. At one point, when she did full emotional/educational testing, we found out she was PG (not a complete surprise, given that she had already been grade skipped after being the youngest in her class to start).
She is now 21, on no meds, set to graduate from college, and doing quite well. I don't really know if any of those diagnoses are correct. I've kind of given up on figuring that part out. She is still intense, still emotional, and occasionally struggles (she took five years for college in part due to emotional stuff, not because of coursework).
I saw a lot of what you are describing - the intense, overwhelming meltdowns, followed by remorse (luckily for us, she was always extremely polite to strangers/school staff; she has a bit of fear regarding authority figures). OTOH, she used to have moments of incredible anger/rudeness/hostility towards us (it didn't help that she went through some really tough emotional stuff - my life-threatening illness, a half-brother passing away, and the death of her dad-although luckily I was remarried before ex's death to a very supportive and loving guy who has been great for her). She is very sensitive still, although now it happens much less frequently. However, right now work and school are both busy and she was unusually frustrated this week and threw up her hands and said she couldn't deal with how "crazy" and "loud" her brothers were being (they're actually pretty quiet and well-behaved). It just seems that her sensitivities are so much greater than most people. Things do NOT roll off her!At least as she's gotten older she can regulate it so much better. Honestly, she also feels sad now for some missed opportunities, mainly to do with college. Although she lived on campus for the first four years, we kept her close at a nearby university due to her age (16)and emotional issues. I think it's hard when you have the potential to be curing cancer (or whatever) and you're just being an "average" (very good) college student, if that makes sense!
I think that you are handling things well - we also learned that threatening/giving consequences/shouting in the middle of any of the meltdowns was hopeless. The most we would do is say "we can't talk to you while you're so upset" - the weird thing is (well maybe not so weird) that she wanted company in a way - hard to describe, but I think it scared her to feel out of control.
We have had a little of the "you like them better than me" (her sister, who is incredibly easy and trouble free) and her brothers (who were younger and she felt "cuter") but I think that came of her own guilt and fears over her behaviors. It wasn't until she was older and more emotionally mature that we really understood her guilt over her actions and how deeply it affected her.
I feel like I'm talking in circles, but I just want to say that I do understand a lot of where you're coming from. The doctor who did the IQ testing basically said that there was nothing wrong with her except her exceptional giftedness. I don't actually believe that, but I do think that it wasn't really about any one diagnosis, but just about trying to help her understand and manage her intensity and how deeply she felt everything.
Okay, still rambling... you may certainly PM me if you want to talk. Hang in there.