It's possible. Sometimes fans come up with creative crossover ideas. But it could take some digging through various online platforms dedicated to fanfiction to find good ones.
Not usually. The Rani is a character from Doctor Who and doesn't commonly crossover into Harry Potter fanfiction.
You might find such fanfic on popular fanfiction websites like Archive of Our Own (AO3) or FanFiction.net. Just use the relevant search terms and browse through the results.
There are all sorts of fanfics out there. Some might have the Doctor visiting the wizarding world of Harry Potter, while others could bring Harry and his friends into the Doctor's time-traveling adventures.
One idea could be a crossover where Doctor Strange uses his magic to visit the wizarding world of Harry Potter. He could meet Harry and help him with some of his more difficult magical problems, like dealing with a particularly powerful dark magic spell that the Hogwarts professors are stumped by.
You could have a story where Doctor Who travels to the magical world of Harry Potter and teams up with a powerful magician to solve a mysterious threat.
Well, you could have a story where Harry Potter travels through time and space with the Doctor to magical worlds and has adventures.
Another possibility is a story in which a young wizard in the Harry Potter world discovers a hidden portal that leads to the dimension where Doctor Strange operates. This wizard, inspired by Strange, begins to study a new form of magic that combines the principles of both universes. They then return to Hogwarts and start to teach their friends this new hybrid magic, which causes quite a stir among the faculty and students alike.
Fanfiction.net has a search function. Just type in 'Harry Potter' and 'Doctor Who' together, and it'll show you the fanfics that combine both. Also, check out the relevant categories or communities for a better selection.
Well, some fanfictions could have Harry Potter using his death powers to assist the Doctor in defeating powerful foes. Or maybe they team up to save different timelines. It really depends on the creativity of the authors!
To find such a fanfic on FanFiction.net, try browsing through the relevant categories like 'Crossovers' or 'Fantasy'. Also, use the site's search function with detailed tags to narrow down the results.