Another theme is the discovery of new magic. With the merging of two different magical concepts, fanfiction often features characters exploring new forms of magic that are unique to this combined universe. For example, spells that are a combination of Hogwarts magic and the powers of the demigods. This makes the stories exciting as readers get to see something new and different.
Yes, there are quite a few. You can find them on various fanfic websites. Some might focus on crossovers between the two universes, like a character from Harry Potter ending up at Camp Half - Blood and having to adapt to the new environment and its rules.
In such a fanfiction, Harry could discover the world of Greek mythology. Maybe he'd meet demigods and learn about their powers. He might also compare his magic from the wizarding world with the powers of the demigods at Camp Half - Blood.
Well, he might be really confused at first. The magic in the Camp Half - Blood is different from what he's used to at Hogwarts. He could make some new friends, like the demigods there. Maybe he'd learn new combat skills too.
One interesting plot could be Harry Potter arriving at Camp Half - Blood and being mistaken for a demigod at first. He then has to prove his own magical abilities which are different from the demigods'. As he stays, he starts to learn about the Greek gods and their ways, and he tries to find a way to combine his Hogwarts magic with the powers and knowledge at the camp. Maybe he could even help the demigods in a battle using his unique spells.
One common theme is the clash of cultures. The magical world of Harry Potter and the demigod world of Camp Half - Blood have different rules, so when characters from both worlds interact, there's often a struggle to understand each other.
Harry Potter could fit into Camp Half - Blood in a fanfic by showing his loyalty and friendship skills. He's known for being a great friend at Hogwarts, and those qualities would be useful at the camp too. Also, his magical abilities, though different from the demigods', could be an asset in certain situations, like when they need to solve a mystery or overcome an obstacle.
You can try fanfiction websites like FanFiction.net or Archive of Our Own (AO3). They have a vast collection of fanfictions, and you might be able to find the 'Harry Potter finds Camp Half - Blood' ones there. Just use their search functions and type in relevant keywords.
If Harry Potter was raised in Camp Half - Blood, he would likely be trained in a different way of magic. The demigod training there focuses on physical combat as well as magic use. He might have an easier time with some magical creatures that are common in the Greek mythology world of Camp Half - Blood compared to the ones in the wizarding world. His wand might not be as crucial as it is in the wizarding world, since demigods rely more on their inborn powers.