When is it acceptable to kill a character? (and if possible, when is it most appropriate?)
BTW, I'm not talking just about a protagonist; I'm talking about any of my characters, whether they're MCs or extras is irrelevant.
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When it serves the plot.
That's really the only reason you do anything in a plot-driven story. If the death serves the plot, do it. If it doesn't serve the plot, then don't.
Don't do it for shock value, or purely to motivate another character (that's called "fridging," which you can read about on TV Tropes), or because it's trendy or edgy or GRRMartin or J Whedon does it.
Do it because there is a reason which makes sense for your plot to move forward.
Lauren Ipsum's point is spot on, but I'd like to elaborate further. Does killing them off or keeping them alive bring better conflict for the story, and deliver a better story arc for other characters than the character who lives instead of dying?
G. R. R. Martin is an obvious author to learn from. Even though he has a habit of killing off major characters, they generally form vital story arcs for other characters that increase overall conflict. A couple of spoilers from Game of Thrones to illustrate this point:
Consider the death of Ned Stark. If he had been spared, Arya Stark's journey motivated by revenge simply would not have occurred. It's questionable as well whether or not Ned Stark could have developed in such a way that it would have served the story in a better fashion and created better conflict. It also conflicted with what we knew about Joffrey's cruelty, and would have been less believable. Even though many people hoped Ned would survive, it was far better for the story that he didn't. It also provided GRR Martin what he ultimately needed, which was to demonstrate to the reader that the heroes of the story are not safe (more on that below)
Consider as well Tyrion's murder of his father. Without this action, the house of Lannister would still have had his father at the head of the household, and there would have been very little change in the relationships between Tyrion and his siblings.
Killing off a major character is exceptionally useful when done right, because the reader always wills for the protagonists to succeed, but now there's the overall danger that they may not. Clearly, no-one is safe. This creates the tension needed for a reader to want to keep reading to see what happens. (This is one of the main reasons many superhero stories can be rather boring; the heroes are so powerful you pretty much know from the outset they're going to survive and win.)
Generally, non-MC characters are killed to provide some sort of motivation or commitment to other characters, or to prove the lethality of the setting and raise the stakes of whatever the MC is doing to life-or-death. One more reason is self-sacrifice, somebody dying for their cause. This is often combined with a love motivation, a parent dies saving their child, a soldier dives on a grenade to save his brothers-in-arms.
Bruce Wayne's parents are killed to justify Bruce's emotions that drive him into a lifelong journey toward becoming a vigilante and preventing others from experiencing his plight. Spiderman has a similar back story.
In such movies, there is often an expression of love for the killed character by the MC in some way, even if it is back-handed love (an argument or disagreement with a sibling over something trivial) to add an element of guilt to the MC about their friend/sibling/parent dying on a sour note in their relationship.
Alternatively, many a villain is motivated by revenge for the death of a loved one.
In Star Wars, Indiana Jones, gang stories, spy stories, cop stories, disease stories, extras and walk-ons are slaughtered by the author with gleeful impunity to reinforce that the MC is in lethal danger. (Often by the MC themselves). The MC is often put in situations that cause them harm, even close to lethal harm, to prove the point. They get shot, stabbed, break an arm.
The "partner killed in the line of duty the day before retirement" has become a laughable cliché, but the first time it was used it probably delivered with good impact.
Another cliché is the walk-on, well-loved, long-lost college buddy that ends up dead by the end of the show, giving the MC motivation to disbelieve the "usual" explanations and dig deeper to prove the death was no accident.
In a show like "Erin Brockovich", the eponymous MC is motivated to extraordinary investigative lengths out of deep sympathy for the ill and families of the dead and dying, even though I don't think she was ever in any imminent danger herself. (It's been a long time, but I don't think she was).
+1 Lauren, "When it serves the plot". These are ways and examples of how it serves the plot, and I will disagree and say the death can be purely to motivate another character, but if that is going to be the case, you probably need to "show, don't tell". Don't just say it in exposition, you need to show some depth of emotional connection to the doomed character by the character you wish to motivate. The motivation, for hero or villain, arises from the severing of some kind of love connection, be it parental (either direction), romantic or platonic (aka non-sexual, including siblings, friends, respected mentors, partners, etc).
Without the love connection, most character deaths in good fiction are examples of lethality, villainous or not. E.g. a fellow cancer patient has a seizure and dies unexpectedly, or in Gravity, a fellow astronaut is killed by high speed debris smashing through their helmet and destroying their head.
Or, the death is self-sacrifice: Obi Wan dies as his only option against Vader that allows him to continue protecting Luke (and it is necessary in the plot for Luke to be forced into autonomy.)