In this kind of fanfiction scenario, Snape might initially be conflicted. On one hand, he has his long - standing dislike for James Potter, Harry's father. But on the other, his sense of duty as a Hogwarts professor could kick in. He might reluctantly start to heal Harry, all the while muttering under his breath about how much trouble Harry always seems to get into, just like his father. Then, as he heals Harry, he might start to see some of Lily, Harry's mother, in him and soften a bit.
One place to start looking for the missing fanfiction could be on large fanfiction platforms like FanFiction.net or Archive of Our Own. They have extensive collections of Harry Potter fanfictions. Check their search functions, maybe filter by characters (Snape and Harry Potter) and relevant tags.
If Snape finds Harry Potter fanfiction, he'd probably be quite irritated. He has a rather stern and no - nonsense personality. He might see it as a frivolous waste of time and be disdainful of the creative liberties often taken in fanfiction. He'd likely scoff at the idea of people writing about his students and the wizarding world in such unregulated ways.
Snape could be very cold at first, but then his natural instinct as a healer (since he is quite good at potions which are often used for healing) might take over. He might give Harry a potion to ease the pain and start to patch him up. All the while, he would probably be scolding Harry for being so reckless and getting into a situation where he got injured.
Snape would likely be angry at first. Harry has always been a source of trouble in his eyes, and running away would just add to that perception. He might mutter some sarcastic remarks under his breath about Potter's irresponsibility.
Look for any unique plot details you remember. If it involves Snape finding Harry Potter in a particular place, like a hidden chamber or a different magical realm, use that as a search keyword. For example, if it was in a 'forbidden forest' setting, search for 'Snape finds Harry Potter in the Forbidden Forest fanfiction'.
In some fanfictions, Snape might use his advanced knowledge of potions to create a special concoction to heal Harry. He could draw on his skills in Herbology as well to find rare ingredients for the potion.
Well, if everyone found out Harry was Snape's son, it would completely shatter the existing relationships. For one, Harry's relationship with the Dursleys might be even more strained as they already dislike him, and this new revelation could make them feel even more threatened. His relationship with Ron and Hermione would also be tested. They might be confused and even a bit betrayed at first. Hermione, being the logical one, would probably try to find out how this could be possible. Ron, on the other hand, might have a more emotional reaction. And as for Snape himself, his whole attitude towards Harry would have to change drastically. He couldn't keep up the pretense of hating Harry when it turns out he's his son. It would also change the way the wizarding world views Harry. Some might be more sympathetic, while others might be suspicious of Snape's motives all along.
Snape might suspect that there is some dark magic at play. Given his knowledge and expertise in the Dark Arts, he would immediately look into any signs of foul play. He could also be conflicted because on one hand, Harry is a student he has had many issues with, but on the other hand, he is also an important figure in the fight against Voldemort. So he would have to decide whether to report Harry's disappearance or try to handle it privately.
In some fanfictions, Harry might take the injured owl to Hagrid. Hagrid, being an expert with magical creatures, would then tend to the owl's wounds. He might use various potions and spells to heal it. For example, he could use a healing charm that he learned from his time at Hogwarts or some special herbs from his garden.