The Volturi's visit to the Cullens in Twilight fanfiction can bring about various scenarios. It could start with a formal greeting but quickly turn into a tense negotiation. Since the Volturi are known for their strict rules, they might question the Cullens about their relationships with humans, like Bella. They may also be interested in the unique abilities of the Cullens. For example, if there's a new power among the Cullens, the Volturi would want to know more about it. And the Cullens would have to be very careful in how they handle this visit to avoid any conflict that could lead to a battle.
The Cullen family members will have different reactions. Carlisle, as the patriarch, would be diplomatic. He would try to smooth things over and show that the Cullens are following the rules. Alice might be using her gift to foresee any potential danger during the visit. Emmett, on the other hand, would be more ready for a fight if it came to that. As for the Volturi, they would come in with an air of authority, and their members like Aro would be observing every little detail of the Cullens.
A major theme is family. The fanfictions usually focus on the bonds between the Cullen family members, how they support and sometimes clash with each other. Also, the idea of immortality and how it affects their lives, relationships, and view of the world is quite common.
No, they probably don't. The Cullens are fictional characters within the Twilight universe. Fanfiction is a creation of the fans in the real world. The Cullens are busy with their own lives in Forks, dealing with things like going to school, hunting, and protecting Bella. They don't have access to the concept of fanfiction as it exists outside their fictional realm.
I'm not sure of specific 'Cullens help the Volturi' fanfics off the top of my head. But you can try searching on fanfic platforms like Archive of Our Own (AO3). They have a large collection of Twilight - related fanfics and you might be able to find some with that particular plot there.
One possible plotline could be that the Volturi face a new and powerful enemy, perhaps a group of vampires with unusual abilities. The Cullens, with their unique skills and more compassionate nature towards humans, decide to help the Volturi in order to maintain the secrecy of the vampire world. They might use their connections with the human world and their knowledge of co - existing with humans to their advantage in this fight.
Well, they could be interested in fanfiction as it might show new and creative storylines. The original Twilight story has its own plot, but fanfiction can take it in various directions. For example, fanfictions might explore what would happen if a Cullen fell in love with a different type of character. This novelty could attract the Cullens to read such stories.
They might be amused. After all, fanfiction often takes their own story and twists it in unexpected ways. It could be like looking at their lives from an outsider's, albeit a very creative outsider's, perspective.