Some fanfictions choose the snake. This is an interesting choice as it is associated with Slytherin and the Dark Arts. Harry turning into a snake could be a way to explore his connection to the darker side of magic, or it could be a plot device to infiltrate the Slytherin common room or the Death Eaters' lairs.
The owl is also popular. Owls are already an important part of the wizarding world as messengers. If Harry turned into an owl, he could travel quickly between places and deliver important messages himself. It gives a new aspect to his role in the story.
Hippogriffs are popular as well. Their majestic appearance and the fact that they require respect to interact with make for interesting stories. Fanfictions might explore a young wizard's journey to tame a hippogriff and the bond that forms between them. There could be a story where a hippogriff is wrongly accused of attacking a wizard, and the main character has to prove its innocence.
Hedwig would be a very useful ally. Since she can fly and is already Harry's pet, she could be his eyes and ears in the sky. She could deliver messages faster and also gather information from other animals she meets during her flights.
Hedwig is likely to be one of the main animal characters. She was Harry's owl and had a big role in the original series, so in the fanfiction she might have even more expanded storylines. Buckbeak the hippogriff could also be a main character. He was already quite important in the books and could have new adventures in the fanfiction.
Harry Potter himself is often written in these types of fanfiction. He can be either the 'little' or the 'caregiver' depending on the story. Hermione is also a popular choice. Her intelligence can be used in different ways in the DDLB relationship, like being a very nurturing 'caregiver' or a 'little' who is learning a great deal. Ron Weasley is another one. His loyalty can be a big part of the relationship dynamics.
Harry Potter himself is obviously still very popular as the main character. His growth and development as the Chosen One are often central to these fanfictions.
Yes, there are such fanfictions. You can find them on various fanfiction websites.
Hermione Granger is often a character that is transformed in these fanfictions. She's a very popular character with a strong personality, so seeing her in a different gender role can lead to all sorts of interesting plot developments.
Harry and Hermione are often swapped. Since Hermione is such a strong and capable character, it's interesting to see her in Harry's role. Also, Neville and Harry are commonly swapped as Neville also has the potential to be a hero.
Harry Potter himself is often included. He's the main hero of the original series, so it makes sense. Hermione Granger is also a common choice. Her intelligence and strong - willed nature make her an interesting addition to a harem. Ron Weasley can be included too, though perhaps less frequently. And then there are characters like Draco Malfoy. His complex personality can add a lot of drama to the harem dynamic.
A hippogriff is also a possibility. In fanfic, if Harry turned into a hippogriff, he would have to learn how to control his new form. He could make friends with other hippogriffs and gain their trust. This could lead to new alliances in the wizarding world. Also, his hippogriff form would be very useful in battles, as he could use his beak and hooves to attack enemies.