Dumbledore could also play a role even if he's no longer alive in the normal sense. His portrait in the headmaster's office might give Harry some cryptic advice about the sword. Or perhaps his spirit could appear to Harry in a moment of need and guide him on how to use the sword effectively in the fight against the forces of evil.
One possible plot could be that Harry stumbles upon the Sword of Gryffindor in a hidden chamber in the Forbidden Forest. Maybe he's on a solo quest to find a powerful weapon against Voldemort and the magic of the forest guides him to it. Then he has to figure out how to use its power to his advantage in the upcoming battles.
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.
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.
Hermione would most likely help. She's always been there for Harry, and with her vast knowledge from books, she could offer a great deal of advice on how to take care of the dragon egg. Ron might also be involved. Despite his initial fear of things, he has a brave heart and would probably stand by Harry's side.
One possible plot development could be that Harry first goes into denial. He has grown up thinking his situation was normal. But as he realizes he's been abused, he might start to distance himself from those who have been part of the abuse, like the Dursleys. He could then seek out help from his friends at Hogwarts, like Hermione and Ron. They would support him, and together they might try to find a way to deal with the emotional and psychological trauma he has endured.
Hermione would surely be one of the first to help. She is highly knowledgeable in magic and would likely try to use a healing spell or find a potion to stop the bleeding.
The genie might be a transformed wizard. Some powerful dark magic could have turned a wizard into a genie as a form of punishment. When Harry finds the lamp, he unknowingly releases this wizard - genie. This genie - wizard could have knowledge of magic that has been long forgotten, which could be very useful for Harry in his fight against Voldemort.
The baby could be the child of a minor character from the original series who was in hiding. For example, the baby might be the offspring of a wizard who was on the run from Voldemort's followers and unfortunately didn't make it, leaving the baby behind for Harry to find.
Hermione would most likely be the first to help. She's always been there for Harry and is very knowledgeable in potions. She might quickly assess the situation and try to find a counter - potion or spell to revive him.