At first, it's likely to be very tense. Snape has always had a grudge against Harry because of his father. But as they live together, they start to see each other's true selves. Harry might see Snape's vulnerability and intelligence, while Snape might see Harry's courage and determination that are different from James's arrogance.
Perhaps in this fanfiction, Harry discovers some secrets about his parents from Snape that Dumbledore had been keeping from him. This could lead to a whole new understanding of the wizarding world's politics and relationships. Snape could be forced to protect Harry not just because of his loyalty to Dumbledore, but also because of his newfound connection with Harry.
The relationship would be a rocky one. Snape has his past grudges against Harry's father. However, living together, they would be forced to interact daily. Harry might start to understand Snape's difficult past and Snape could see Harry's potential. There could be moments of anger, but also moments of mutual understanding, like when Harry helps Snape with some magical experiment.
One possibility could be that Dumbledore arranges for Harry to live with Snape for protection. Snape, though reluctant at first, gradually discovers Harry's hidden strengths and they form an unlikely bond. Harry might learn more about his mother from Snape, which could lead to a deeper understanding between them.
I really like the fanfic 'Snape's Second Chance'. In this story, Snape gets to correct some of his past mistakes. He has the opportunity to build better relationships with the students at Hogwarts. Also, 'Living Snape, Loving Snape' is a unique take where he finds unexpected allies and forms new friendships. It's quite heart - warming to see his character develop in these fanfictions.
One common theme is redemption. Since Snape has had a rather complex and often misjudged past, many fanfictions explore his chance at redemption in his new - found life. For example, he might try to make amends with characters he wronged or prove his loyalty more clearly.
Sure. Knockturn Alley is a place full of mystery and danger in the Harry Potter universe. For a fanfiction writer, it's an exciting setting to place a character like Potter Snape. He could have all sorts of adventures there, from getting involved in illegal potion - making operations to trying to clean up the alley's more unsavory elements.
At first, it would be really rocky. Harry would be suspicious of Snape and Snape would be his usual grumpy self. But over time, as they get to know each other better, they might start to respect each other. Harry could learn about Snape's past and his complex relationship with his mother, which might soften his view of Snape.
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.
In some fanfictions, Snape might use his advanced knowledge of potions to brew a special healing elixir for Harry. He could also use his magic skills, like a complex spell specifically designed to mend Harry's wounds, whether they are physical or emotional.