Sirius Black might be involved. As Harry's godfather and James' friend, he might have some knowledge about the trunk. He could help Harry understand the significance of the items in it. Also, Dumbledore could be involved. He always seemed to know a great deal about everything, and he might have some information about Lily's trunk that could help Harry on his journey of discovery.
Petunia could also be involved in this fanfiction. Even though she was estranged from Lily, there could be a backstory related to the trunk that involves her. Maybe she was jealous of something in the trunk when they were younger. And Hagrid might play a role too. He was always around Hogwarts and could have some memories or knowledge about Lily's school days and the trunk.
In the 'Harry finds Lily's school trunk' fanfiction, it could be that Harry stumbles upon the trunk and inside he finds a diary. Through this diary, he gets to read about Lily's adventures at Hogwarts, her friendships, and her early encounters with Snape. This would be a great way to explore Lily's character more fully from Harry's perspective.
Perhaps Sirius Black could also be involved. Sirius was Harry's godfather and was very much a part of Harry's life. His presence could add some interesting dynamics, like he could be the one who stumbles upon the magic that makes the meeting possible and then has to deal with the implications of Harry meeting his mother.
There could be her personal diaries. Maybe she wrote about her time at Hogwarts, her friendships, and her early days with James Potter.
Even some of the professors could be involved. Professor Dumbledore, with his vast knowledge and wisdom, might have some connection to the magical trunk. He could be the one guiding the students in their exploration of it, or perhaps he has left some clues for them to discover. Professor McGonagall might also get involved if she senses that the students are in over their heads with the trunk's secrets and she needs to step in to protect them.
Dumbledore could be involved. He often has his fingers in many pies at Hogwarts, so he might be the one who arranges for Snape to deliver the letter to Harry for some greater good or part of his grand plan. Another possible character could be Hermione. She might notice something strange about Snape delivering the letter and start to investigate, suspecting that there's more to it than meets the eye.
The teachers, such as Professor McGonagall or Professor Snape, would most likely be involved. McGonagall could be the one administering the test, while Snape might be observing with his usual critical eye. Also, other students in his year could be taking the test alongside Harry. They could be either competing with him or collaborating in some parts of the test. This would add an element of drama and friendship or rivalry to the fanfic.
James Potter often makes an appearance. Since he was Lily's husband, his relationship with both Harry and Lily is explored in different ways. For example, in some fanfictions, Harry gets to see a side of his father through Lily's memories that he didn't know before.
Yes, there are likely to be such fanfictions in the vast world of fanfiction. Many fans like to create alternate family relationships and storylines for their favorite characters.
Hermione and Ron would most likely be main characters. They are Harry's closest friends in the original story. Since Harry is deaged, they would play important roles in taking care of him or guiding him in some way.
Her best friend could be a main character. This friend might be the first one Lily tells about getting her period and could offer practical advice like which pads are the most comfortable.