I will be using profanity and language some might find offensive in this post, including racial slurs for the purpose of example. Deal with it or don't read it.
What sorts of things are ok to censor and why?
Acts and statements made in a public space that advocate illegal action, especially violent action, except when done in an artistic or entertainment medium. If someone wants to talk about how niggers are awful, okay, they're a dick, but free speech is in play; if they instead say niggers are awful and need to be killed like dogs in the street, yeahhhh, censoring that is fine. For the same exact reasons I would also say it's fine to censor people who tell others to go pirate things. Anything that could directly and logically influence people to go commit crimes is reasonable to censor, and again this is especially true for advocating violence. It's not okay to encourage crime, those things are illegal for a reason, so censoring that shit is okay by me.
I will, however, add two exceptions to this. First, some things are illegal but honestly who gives a shit if you tell people to do it, like jaywalking, so those can slide by; when I say "crime" and "illegal action" I mean actual serious crimes that do some kind of harm to other people and have actual jail time sentences attached to them. Second, when it comes to historically important items of hate speech advocating violence (such as the speeches and writings of Hitler and various other shitbags), I feel that it is very important that they not only remain uncensored but that people be exposed to them so they can understand what sort of horrid things have happened in our history. As the saying goes, those who don't learn from the mistakes of history are doomed to repeat them. You can't really teach people about Hitler and why he was so awful and also censor his awfulness, else it loses its impact and makes the lesson unimportant.
That's pretty much it. That's the only thing I can think of that is okay to be blanket censored.
Should minors have more censored from them than adults? Is censoring children futile with the accessibility of the Internet?
Eh, it's ultimately futile, but it's also reasonable to censor certain things from minors. Mainly the youngest of minors, like prepubescent kids. There's no reason to show young children a graphic sex scene, for example. Censoring graphic violence is also okay for young kids. It's not likely to actually fuck them up unless you show it to them before they can fully distinguish the difference between reality and fiction, but even after that point why would you show a gory slasher film to like a 10 year old? They almost universally don't have the mental development to be able to process those things in a reasonable manner, even knowing it's fiction.
But honestly, once they get past puberty it's basically pointless. Even before the internet existed the older age bracket of minors was pretty persistent and crafty in viewing that sex and violence that adults tried to keep away from them. There's a natural curiosity there, especially as it pertains to sex, and barring their access to it only makes it more influential to them when they finally get hold of it (because instead of just being another thing that exists it's now built up in their mind as a titillating taboo). Let the ~13 year old kids watch bloody murder movies and stuff with not-very-graphic sex scenes in it, whatever, it's fine. It'd be reasonable to match age-gated pornography to the local age of consent, but again, basically futile thanks the internet and hormones and developing libidos.
In what context is something ok to be censored? Is it not ok in another context?
See my first and second answers for a full answer of what context it's okay to censor things in.
It is not okay to censor art and entertainment, aside from the youth exceptions above, in almost any way or context. If something is blatantly and clearly calling for the exterminations of the Jews and telling people to go and murder all Jews they can find, with only a thin veneer of artistic shit over it to try to make it acceptable, okay go ahead and censor it. If instead it's something like a World War II era movie or something with a Hitler-esque character and includes character speeches that say stuff like "kill all the kike scum" and whatnot, something that is clearly a piece of art or entertainment that is not actually advocating such action but just contains a character who thinks that way, it is not okay to censor that. I don't care if there's a character that is literally screaming and ranting for minutes on end about how the Zionist machine is secretly oppressing the world and the only way to break free is to finish what Hitler started, as long as it's clearly the character within the piece of fiction supporting these statements with no intent to make real people think the same things then it should remain wholly uncensored.
In a related but broader context, it is not okay to censor something just because some particular group is offended by it. There's a very popular Stephen Fry quote that has gone around the internet attached to images of his face that explains this quite well: "It's now very common to hear people say, 'I'm rather offended by that.' As if that gives them certain rights. It's actually nothing more... than a whine. 'I find that offensive.' It has no meaning; it has no purpose; it has no reason to be respected as a phrase. 'I am offended by that.' Well, so fucking what." There's also a related but more succinct version of the same sentiment that comes from Ricky Gervais: "No one has the right not to be offended, and don't forget, just because you're offended doesn't mean you're in the right."
Everyone has the right to be offended if they so feel the need, but at the same time everyone should have the right to express their thoughts however they would like so far as they are not breaking laws or inciting law breaking. If you don't like that someone said something offensive, stop talking to them or viewing their art/entertainment, don't try to fucking censor them because you're upset. That's bullshit. I don't care what kind of vile filth someone is spewing, so long as they are not causing real physical harm, breaking laws, or telling others to do so, then it should not be censored. I just remembered another quote that relates to the matter, often falsely attributed to Voltaire when it was actually first said by Evelyn Beatrice Hall in a biography to paraphrase Voltaire's stance on a matter: "I disapprove of what you say, but I will defend to the death your right to say it." That's one of the basic principles of freedom of speech, and I wholeheartedly agree with it.
Is the current system of what does/doesn't get censored in our media, and who chooses whether or not to censor it fair? If not, what could be done to fix it?
Fair? Not particularly. I'll focus on movies and video games here because those are the two areas of media where I actually know a good deal about the censorship involved. No idea what goes on with music, television, book, or other media censorship at more than a very general level; on that general level, I don't like censorship, so it's fair to assume I'm not okay with any of it that's done to mediums other than video games or movies unless it falls under my already listed exceptions.
In the US, censorship of media is often done through soft but commercially crippling methods rather than actually blocking certain things from being viewed. They slap ratings on media that act as age gates, and while these ratings are not actually enforced by federal law, they are pretty effectively set in stone because media providers will bar people from watching or buying something that is rated to say they are too young for it. These are not government-appointed groups, they are organizations formed by the industries themselves to regulate their products. The Motion Picture Association of America was formed by the big six Hollywood movies studios back in 1922, and it's been in the business of censoring film content since 1930, which is pretty strictly enforced in theaters around the country despite there being no laws in most places saying that young children cannot watch R-rated movies. The Entertainment Software Rating Board was formed by four of the big game developers and publishers in response to Congress considering making a formal legal body to regulate video games after public outcry about violent and sexual content present in some of them (particularly Mortal Kombat and Night Trap); just like with film ratings, places that sell video games will almost never sell them to people younger than the rating says is okay.
So, you've got these self-regulating bodies formed from within the industry itself that decide what ratings to put on games. On the face that might seem great, because it's not the government censoring media, but in truth it's just a less direct kind of censorship. Instead of saying "no, you cannot do that," if they find something objectionable they say "okay, we'll give you the highest level content rating." What does that mean in technical terms? The top two ratings for movies are R ("People under 17 years may only be admitted if accompanied by a parent or guardian") and NC-17 ("This film is exclusively adult in content and people under 18 are not admitted"); for video games it's M/Mature ("Games with this rating contain content which the board believes is suitable for those aged 17 years and older") and AO/Adults Only ("Games with this rating contain content which the board believes is unsuitable for people under 18 years of age"); so the technical difference is basically one year of age. What does this mean in practical terms? Theaters and video game sellers will often flat out refuse to show or stock anything with the highest rating because it's inappropriate, thus it would mean that the product will be almost guaranteed to flop and not make much money. This forces the content producers, which are businesses first and foremost so they want to make money on their products, to cut out content the ratings boards don't like until they'll agree to drop it down to the second highest rating and back into commercial viability. This is absolutely and unequivocally a form of censorship.
What's worse is that the lines between the various ratings levels are quite stupid. Movies could have no violence or sex, but they have a character say "fuck" a few times so it gets slapped with a PG-13 rating. Some movies can be full of violence and get away with a PG-13 rating, but cut out most of the violence and throw in a romantic subplot that includes a sex scene and might instead be rated R. It's similar for video games: a game with very mild and cartoony violence and no cursing could be rated E/Everyone, but add just one joke about boobs and you're probably getting a T/Teen rating for crude humor. It's ridiculous, especially with video games. For example, Grand Theft Auto: San Andreas was temporarily rated AO because if you modified the game (meaning it was not accessible in normal game play at all) you could access a totally non-graphic sex minigame that just showed poorly rendered models of the player character and various girls getting into sexual positions and making the motions with no genitals or anything actually visible; what makes this ridiculous is the fact that the game got by just fine with an M rating when it had basically the same exact sound files accessible through normal play (just moans and goofy things they say), plus extreme violence like beating people to death with a rubber dildo or getting a tank and blowing fucking everything up, because apparently gratuitous murder is less offensive than clothed character models dry humping. Sex and anything pertaining to it is absolutely awful and totally offensive as far as these ratings boards are concerned, which I find to be utterly ridiculous.
The system could be improved by changing the rating systems. Make a "family friendly" rating and a "contains mature content" rating, where the split is stuff you'd be fine with 8 year old kids (or whatever age you want, young children, basically keep this as what the PG/E ratings are today) viewing and everything else in the group for mature content. Include a list of what specific items of mature content are present in each thing, rather than saying "oh, you hit seven sections of mature content, you get an M/R rating instead of a T/PG-13." Then you let businesses decide just which things they'll actually stock, instead of the whole "omg no, can't have the top rating stuff because that's evil and wrong!!!" thing that goes on currently. Sure, they still won't want to provide pornography to minors, but at least it'd make it so each piece of media needs to actually be looked at with a critical eye by both providers and consumers to decide what they want to do with it. This would be good for lessening censorship of content for minors as well: let parents actually see that what would be a scary M rated game just has extreme cartoon violence, alcohol use, and crude humor instead of thinking it's basically porn or murderporn, and they can decide based on the actual content rather than the broad rating whether or not their kid can have it. Much better than allowing these self-regulating groups to censor shit with rating threats.
Are warning labels sufficient substitutes for outright censoring?
Absolutely. As per my above suggestion for how to fix media censorship in the US, I think that instead of outright censoring things you should provide a list of what content exists in a piece of media. Let people decide whether or not they want to view the thing based on its content instead of blindly following ratings. It could also be quite nice for people who have specific things that they're not okay with viewing. For instance, someone who has been raped or someone who was abused as a child could see that a movie that would be just an R rating under the current system for sexual content and violence has a rape scene or contains child abuse scenes and decide to avoid it because they don't want to see such things. Much better than the current system of "OMG THAT SAYS ADULTS ONLY THEREFORE IT IS AWFUL AND MORALLY REPREHENSIBLE!!!"
TL;DR fuck censorship right in the ass