Well, it depends on the specific fanfiction. There are countless fanfictions out there, and while some might make Harry the heir of Salazar Slytherin, others may not follow this concept at all. Some fanfic authors like to subvert the expectations set by the original books and come up with unique ideas like this to explore different aspects of the wizarding world.
I read a fanfiction once where Harry being the heir of Slytherin led to a whole new dynamic in the wizarding world. The story explored how the other houses reacted, especially Gryffindor. It was fascinating to see Harry struggle with the duality of being seen as both a hero from Gryffindor and the heir of the supposedly 'dark' Slytherin.
One interesting fanfiction could be 'The Heir of Slytherin's Destiny'. It delves deep into how Harry discovers his heritage as Slytherin's heir and how it changes his relationships at Hogwarts. His friends initially have doubts, but as they see Harry struggle to use this new - found knowledge for good, they come around.
One consequence could be a change in how the other characters view Harry. The Gryffindors might be more suspicious of him, while the Slytherins could either try to use him or see him as a threat to their own status. For example, Draco Malfoy's relationship with Harry would likely be even more complex.
One common theme is identity crisis. Harry has always identified as a Gryffindor, so finding out he's Slytherin's heir makes him question who he really is. Another theme is prejudice. People in the wizarding world often have negative views of Slytherin, so Harry has to deal with that when his heritage is revealed.
Yes, in certain fanfiction. This concept allows for a really interesting exploration of the Harry Potter universe. By making Harry the true heir of Slytherin, it can add a whole new layer of complexity to his character. He could inherit Slytherin - like traits such as cunning and ambition, which would contrast with his Gryffindor - associated qualities like bravery, leading to some great internal and external conflicts in the story.
In some fanfictions, yes. Writers like to play with the idea of subverting the original story. They might create a scenario where Harry discovers he has some hidden connection to Slytherin. It could be through some convoluted family history or a magical revelation.
In some fanfictions, Harry Potter can be made the heir of Slytherin. Fanfiction allows for creative reinterpretations of the original story. Writers might explore this idea to add new twists and turns to the plot. For example, it could create interesting dynamics with the other characters, especially those from Slytherin house.