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.
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.
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.
Well, in Hunger Games fanfiction, Prim is sometimes depicted as even more resilient and influential, having a greater impact on the events of the fictional world.
Don't forget the small gestures. Katniss might bring back a small flower or a trinket from the woods for Prim. And Prim, in return, would take care of Katniss when she's injured or tired. In fanfiction, describe these tender moments in detail. For instance, when Katniss comes back from a long day of hunting with a sprained ankle, Prim gently cleans and binds it while Katniss tells her about her day in the woods.