Dumbledore might also play a role. He has vast knowledge and experience. He could offer some wise advice or even provide a special magical item or spell that is crucial for the unicorn's healing.
In such a fanfiction, perhaps Harry uses his magic skills learned at Hogwarts. He might first find the injured unicorn in the Forbidden Forest. Then, using a spell like 'Episkey' or some advanced healing magic he's discovered. The unicorn could be injured by dark creatures or a trap set by poachers.
His friends from Hogwarts, like Ron and Hermione, are likely to help. Ron could bring his family's practical knowledge of wizarding homes and Hermione would contribute with her extensive research skills.
Hermione would be a great help. Her vast knowledge of magic and books could provide the necessary information for building the planet, like the right spells and magical theories.
Dumbledore might offer some guidance. He has vast knowledge and experience. He could tell Harry about the significance of different elements in a staff and how to combine them to make a truly powerful and unique magical tool.
Dumbledore might help in a spiritual or advisory way. Even though he's passed away, his wisdom could be passed on through memories or some magical means. Also, Hagrid could be a great help. He could assist in building areas suitable for magical creatures with his love and understanding of them.
His best friends Hermione and Ron would surely be there to help. Hermione with her vast knowledge of magic could research about the egg, and Ron with his loyalty would stand by Harry's side in any danger related to the egg.
It's possible that an old acquaintance from Hogwarts who has family ties in America could assist him. This person might already have a network in America and could introduce Harry to important people. Also, there could be some reclusive magical scholars in America who, out of curiosity about Harry's powers and his role in the UK, decide to help him. They could teach him new magical skills that are specific to the American wizarding world.
Ron might be there too. He could use his skills in handling magical creatures to populate the island with friendly ones. And he could also help with fishing in the surrounding waters using magic, providing food for those on the island.
Hagrid would be a likely helper. He has a great knowledge of magical creatures and would surely know a lot about phoenix eggs. He could give Harry advice on how to take care of it.
Dumbledore could also be involved. He's extremely knowledgeable and powerful. He might have some secret knowledge about basilisk eggs. He could guide Harry on whether to keep the egg or destroy it, depending on the greater good of the wizarding world.