Yes, it is a somewhat common theme in some fanfictions. One reason could be the complex relationship between Harry and Snape in the original series. Snape was often cruel to Harry, so in some alternate universes, writers explore the extreme scenario of Harry killing Snape. It also allows for exploring darker aspects of Harry's character and the consequences of such an act within the wizarding world.
Perhaps in the fanfic, Snape discovers a powerful and dangerous magic that he wants to use for what he believes is the greater good, but Harry sees it as a threat. Harry has always been wary of Snape, and this new discovery could be the tipping point. Also, if Snape were to make a deal with a dark force in order to gain knowledge to defeat Voldemort but Harry misinterprets his intentions, it could lead to Harry killing Snape.
Well, in this kind of fanfiction, it might be that the writer wanted to subvert the typical relationship between Lily and Snape. Lily is often seen as a positive figure in relation to Snape. But if she kills him, it could be part of a story where the power dynamics or the moral compass in the Harry Potter universe is flipped. For example, if Lily was influenced by dark magic or some external force that made her see Snape as an enemy when he might not be in the original context.
One plot could be that Snape is discovered to be working on a new dark magic spell that would endanger the whole wizarding world, and Harry, in an attempt to stop him, ends up killing him. Another could be a case of misinformation. For example, Harry is tricked into believing that Snape killed someone he loves dearly, like Ginny, and in his rage, he kills Snape before finding out the truth.
In some fanfictions, Snape kills Dumbledore as part of a greater plan. Maybe Dumbledore was already dying or had some sort of incurable dark magic on him, and Snape's act was a mercy killing. It could also be a complex plot twist where Snape is actually trying to protect the wizarding world from a greater threat that only he and Dumbledore knew about.
I'm not really aware of any specifically popular ones. Fanfiction is a vast world and it can be hard to single out popular ones with such a specific and rather dark premise.
Well, it completely flips the dynamic. In the original series, Snape was a complex character with a lot of hidden motives that were ultimately revealed to be on the side of good. If Harry kills Snape in the fanfic, it makes Harry seem more impulsive and less of the heroic figure we know. It also undermines the redemption arc that Snape had in the original story.
One consequence could be that he becomes a hero among his peers. Everyone in Hogwarts might look up to him even more. The other students could see him as a real dragon - slayer, which gives him more popularity and respect.
There could be. Fanfiction is a platform for fans to experiment with different plot ideas. Since there is a lot of tension between Harry and Lucius in the original books, some writers may have been intrigued by the idea of Harry killing Lucius. You can search on popular fanfiction websites to see if such a story exists.
Another way Snape could comfort Harry is by standing up for him in certain situations. Maybe when other teachers or students are being unfair to Harry, Snape steps in and defends him. This act of protection can be a great source of comfort for Harry, as it shows that someone is on his side even if they have had a rocky relationship in the past.