I'm sorry to hear you're dealing with this. Our oldest dd had some similar tendencies and it was very stressful at times for both her and us. She is now in her 20s and we've talked from time to time about her younger self and she describes being completely overwhelmed by her feelings, feeling out-of-control, and being scared (by the intensity of her feelings), but at the same time being unable to respond rationally. I think being a perfectionist is pretty typical for gifted kids, but I think the question that you need to ask is not whether it's typical, but whether it's healthy. In the case of your ds, it sounds like it's not healthy and some intervention is needed. For our dd, part of the resolution was, of course, just simply maturing. However, she was also helped by CBT and by neurofeedback, as well as traditional counseling. What helped from us was to put the brakes on the response to the extent that we were able. She later said that since she felt out of control, the more "in control" that we seemed the better. In essence, we tried:
1) always remaining calm
2) setting boundaries around activities that promoted the behavior (in other words, if he is sitting at the piano getting more frustrated every minute, you need to step in and say he's done).
3) setting clear expectations for acceptable behavior and letting him know what is appropriate. I'm not saying to punish him by any means, but I do think it's appropriate to explain that his behavior is disruptive and scary and not healthy for him, so you are going to step in.
4) Don't spend a lot of time responding or interacting with him when he is out of control, because honestly, you will be frustrated and he won't hear a word you're saying.

I would apply these same things to your dd. It's good she gets over her anger quickly, but exploding, crumpling things up and storming off is also not acceptable behavior. Believe me, as a mom to a very emotionally intense dd, I understand that they are not "misbehaving" per se, but I do think that they need to understand that it isn't an appropriate response. My dd eventually learned to pinpoint the very early signs that she was getting angered or agitated and that made all the difference. We spent a lot of time talking about learning how to do that, helping her journal feelings, etc.

I do not in any way mean to make light of your son's feelings about death, but I will say that we occasionally heard the same things (and obviously responded very seriously). I think it's not completely uncommon among the very gifted kids; they just feel things so intensely. We had more than one conversation about death, suicide, depression, etc.. We made very, very sure that she understood that we needed to ascertain if she was serious and we would not blow off any such statements, and we also made sure she had resources and help. Eventually, the storms calmed, but to be frank, they lasted through HS and it took her a long time to not feel like she was a complete failure if things didn't come easily, the very first time, with no mistakes (oddly, she was the most empathetic of all my kids with anyone ELSE, but never with herself!).

Good luck to both of you. I'm sure you are both exhausted and emotionally overwhelmed by the time it ends (although dd often felt better, lol, once she got it all out and would sometimes wake up the next morning refreshed, while I would wake up with dark circles and feel like I hadn't slept at all, as I tried to figure out what to do).