^ In my point of view, when you shoot someone and you're shocked that he's still alive, it does mean you intended to kill him.
And to me, she looks like she's shooting from anger, not fear, even saying "Damn little brat!!!", not "freaking monster" or anything like that.
Doesn't look scared to me, just lashing out in anger like Bulma usually vengefully does, and still recently.
Then, "Why isn't he dead?" - whether said in shock or not - clearly states what you intended to do to a KID whose words clearly show he misunderstands what you are.
Bulma could have willingly been a kid murderer, not in self-defense out of panic, shivering in fear, saying "don't come any closer" and shooting because being forced to at the last moment, but just because "damn little brat, take this!!".
Unforgivable in reality, no matter if a kid lifts your car and says you're both gonna fight now.
Yet guess what, the manga/anime lense is valid for gags as well as serious moments, which makes this moment perfectly fine, just like casually living with Vegeta - who is not making a mess - after all his murders is perfectly fine.
You just forgive all that in those fictions and magically forget the past, as the characters seem to do. You just forgive bad guys for attempted murders and actual murders they've done in the past when they become good guys in shonen-type anime/manga worlds, whether gag or not.
You wanna punish Vegeta like how he would be punished in reality? Fine, then also punish Bulma for trying to murder a kid instead of talking it through, Kame Sennin for his pervert touchings of girls without their consents, Goku too while you're at it since he endangers entire universes out of selfish desires and is litterally playing russian roulette with lives, Piccolo for beating up a kid on a daily basis, Beerus for exterminating people not when it is deserved but when he's unhappy with the way he's treated even when the exterminated people counts billions of innocents...
As for his "sudden" evolution...
Vegeta is forced to live on Earth, has a clear new objective of seeing the Super Saiyan and surpassing him because he always considered himself as the best Saiyan so far, even obsessing over being the Super Saiyan on Namek to a point that annoyed Freeza. In the meantime spending years trying to reach that objective, he can't stand the fact that this objective he reached keeps getting canceled, so he fights to reach it again and does it, and then it's still taken away. Yeah, he develops an obsession from that point on, that keeps him going just like some people hold on to their dream objective for life, the classic carrot in front of the donkey that keeps getting away when you think you're getting close to it. He stops wanting to murder because it would be inconvenient, then gets used to the fact that it's nice not to murder and to have people around you to interact with. And it takes a whole arc to get there, since he starts by not even caring about saving the lives of his son and wife (meaning even though he has better things to do than kill, he doesn't value other lives anyway), then learns to be proud of his (future) son as a warrior (calling him his more-or-less equal and boasting about it to Cell), then discovers true attachement for the first time once his (future) son gets murdered, then in the next years treats his (present) son right because of all that and recognizes his attachement more and more to a point that ends up leaving him in a crisis of identity, which leaves him ceasing the opportunity to get back to his roots because this change is just too much and he feels like it has softened him down...
I'm not a much of a fan of Vegeta, but I can recognize that for DB level of depth, he's actually one of the characters for which his evolution is the most developed (and more shown through events over time rather than explicitely stated) and his psychology is the most coherent (simple and driven by an obsession is a valid psychology that isn't bad by definition), taking the time it needs to make him very gradually change. It is less sudden than Piccolo going from "conquering the world" to having forgotten about it in a year, and it uses the same kind of "afraid to lose the kid" trauma trigger to finally go through the main threshold of his evolution.
Now before saying I interpret the manga in what YOU call "the wrong way", are you humble enough to question your own interpretations for a bit or are you arrogant enough that you dare to call your interpretations correct by default and mine wrong by default?
At least, I'll leave this conversation having expressed my point of view while having read yours considering it as - of course - a valid and interesting interpretation even though I don't necessarily agree with it, and unlike you, not having the audacity to say that you're something along the lines of "just another deluded one who sees what he wants despite being completely wrong, can't believe such guys are out there" or something like that.
Don't know if that's what you aimed for, but that's the kind of thing that can rub a life-long fan that has followed the franchise in every aspect for almost 30 years the wrong way, just casually being lowered to "just another non-sensical guy who gets it in such wrong ways and doesn't know what he's talking about".
Pretty much done with this thread anyway, as I've seen lots of interesting replies (including yours) and said what I had to say.
So say what you want, or not, I won't check it out either way at this point. Got better things to do than to hold pointless grudges for what might have just been a one-time, clumsy outburst, so see you later in other threads, hoping I won't get the dismissive "another deluded one" treatment later on.