Yes, she might. Ino is a character from Naruto, and many fans of the series enjoy fanfic. Since she has a distinct personality in the original show, it's possible she could be a character that some Naruto fanfic writers focus on, which might make her an interesting character for those who like Naruto fanfic to like in that context.
Yes, in some fanfics Naruto might be depicted as having an affinity for the moon. It could be part of a unique plot device, perhaps related to some new power or a deep emotional connection within the alternate universe of the fanfic.
She might like it because it offers new perspectives on the Naruto world she knows so well. Fanfic can explore different relationships and storylines that the official series doesn't cover.
In fanfiction, the possibilities are endless. Naruto liking Ino can be a really interesting concept. It could be due to the way the author wants to show a different side of Naruto's personality. Maybe in these fanfictions, Naruto sees something in Ino that he didn't notice before, like her intelligence or her kindness. This newfound admiration could then turn into like or even love in the fictional world of the fanfiction.
It depends on the specific fanfiction. Some fanfictions might take a very dark and uncanonical turn where they depict Naruto as having a liking for killing, but this goes against the essence of the Naruto we know from the anime and manga. His character is mainly focused on positive values like redemption and unity.
It could also be that Naruto fanfiction sometimes gives the Kyuubi more agency or a different perspective. In the original story, the Kyuubi is often seen as a force to be tamed, but in fanfiction, it can be portrayed in more complex and interesting ways. For example, stories where the Kyuubi has its own motives and goals apart from just being a power source for Naruto. This added depth might be appealing to it.
He might like it because it's a form of escapism. His life as a ninja is full of challenges and responsibilities. Painting fanfiction allows him to enter a world of his own creation where he can have more control.
The fanfic may also show Naruto with a more sardonic sense of humor, which is a characteristic of the Joker. Instead of his usual positive and sometimes naive outlook, he could be more cynical. This could change the way he interacts with his friends and enemies alike. For instance, his banter with Sasuke might become more edged and full of hidden meanings, much like the Joker's conversations with Batman, which are full of mind games.
Naruto could show his 'like Kakashi' traits in his mentoring style in the fanfic. Kakashi is a great mentor to his students, and if Naruto is like him, he might start to take on apprentices and teach them in a patient and knowledgeable way, passing on not just ninja skills but also important life lessons.