Yes, there's no reason why they shouldn't be allowed. Fanfiction is usually publicly available and as long as they are following the rules of the platforms where it's hosted, they can read it.
It could expand their imagination. Fanfiction often has unique plotlines that can inspire them to think beyond what they've read in the original books.
Annabeth could also be angry, not at Percy but at the situation or the forces that led to his death. She might vow revenge and start on a journey to find out who or what was responsible for Percy's sacrifice and make them pay. This could lead her into a whole new set of adventures and challenges as she tries to come to terms with her loss.
Maybe an antagonist who wants to disrupt the group dynamic. If Annabeth is a key part of a group, punching her could cause chaos and confusion among her friends. It could also be someone who has been influenced by dark magic or some sort of mind - control in the fanfiction. This would make them act against their normal nature and punch Annabeth without really having a personal grudge.
They could become more engaged with the overall story universe. Reading fanfiction can make them think about the original story from new angles. Maybe they start to notice things in the original books that they hadn't before. They might also get inspired to write their own fanfiction or share their thoughts about the fanfiction they read with other friends who are also into the Annabeth - related stories.
Well, some of them could be quite critical. They know the original story well, so if the fanfiction doesn't stay true to the essence of the characters or the overall theme, they might not like it. On the other hand, if it's really well - written and adds interesting twists, they could be very into it. They might also compare the fanfiction to their own ideas of how the story should continue or how the characters should interact.
Well, it depends on who is flirting with her in the fanfiction. If it's someone she doesn't like, she'll probably be dismissive and give cold responses. However, if it's someone she has a bit of an interest in, she could be coy and a bit shy while still maintaining her confident facade.