Well, it's hard to say exactly. Maybe in a fanfic titled 'Survival Together' they do. Fanfic creators can come up with all kinds of scenarios for their stories.
Yes, there are some. You can try searching on popular fanfiction platforms like Archive of Our Own. There might be creative interpretations of a relationship between Gale and Prim, although it's not a typical pairing in the original work. Writers often explore different 'what - if' scenarios in fanfiction.
Archive of Our Own (AO3) is a great place to start. It has a large collection of fanfiction from various fandoms, so there's a chance you'll find Gale and Prim love fanfiction there. Just use the search function and type in relevant keywords.
In Hunger Games fanfic, Prim can play various roles. Sometimes she's a symbol of innocence and hope, much like in the original story. For example, she might be the motivation for the main character to fight against the Capitol's cruel regime.
One possible plot could be that Prim uses her knowledge of herbs and medicine to create traps that either incapacitate or mislead other tributes. She could also form alliances based on healing others in return for protection. Another plot might be that she finds a hidden area in the arena that has abundant food and water sources, and she manages to defend it against other tributes.
When it comes to Hunger Games fanfic about Peeta and Gale, a lot depends on the writer's perspective. Some might write Peeta as the underdog who finally gets his happy ending, while Gale is the one who loses out on Katniss. Others may explore a world where Peeta and Gale become friends, despite their differences. They could bond over their shared experiences in the Hunger Games, like the loss of friends and the struggle against the Capitol. This can make for some really interesting fanfic storylines.
There are several factors that could make a 'Prim wins the Hunger Games fanfic' really interesting. For example, it could explore how Prim's character develops in a more assertive and survival - oriented way. Maybe she uses her medical knowledge in unique ways during the games. In terms of finding such a fanfic, apart from the aforementioned websites, you could also ask in Hunger Games - related Facebook groups or Reddit communities. Some fans might be able to point you directly to a great story where Prim comes out on top.
Some fanfics might depict Prim as a more complex character emotionally. While in the original she was mainly seen as pure and innocent, in fanfic she could have inner turmoil and doubts. For example, she might struggle with feelings of guilt for being the reason Katniss had to enter the Hunger Games in the first place.
Gale would likely become more hardened. The pain of being whipped would make him more determined to fight against the Capitol. He might become less trusting of others, fearing that they could be informants who led to his whipping. For example, he could be more cautious around people he used to consider friends.
In many fanfictions, Prim's location can vary depending on the plot the author creates. Sometimes she might be in a safe haven, while in others she could be in a dangerous situation.
Well, Prim in Hunger Games fanfiction could be anything the author decides. She could have new adventures, face different challenges, or have relationships that weren't shown in the original canon.