Well, Merlin and Arthur's relationship could become more complex in their view. If the fanfiction shows a deeper emotional bond between them than what they thought was there, it could make them re - evaluate their friendship.
For Morgana, seeing how she is depicted in fanfiction might make her realize how others perceive her actions. If she is shown as more sympathetic in some fanfictions, it could potentially change her view of her own place in the story and her relationship with the other characters. She might see new possibilities for alliances or reconciliations.
Some relationships might seem stronger in fanfiction. For example, the bond between Lelouch and his sister Nunnally could be intensified in certain fanfictions. This could make the characters themselves more aware of the potential depth of their relationships.
Ariel and Ursula's relationship could be seen in a new light. If the fanfiction shows Ursula having a more sympathetic backstory, Ariel might start to feel a bit of pity for her, which could change their dynamic from pure antagonism. Prince Eric's relationship with Ariel might also be affected. If the fanfiction has a different version of how they met or fell in love, it could either strengthen or weaken their bond in the eyes of the viewers, and the characters themselves might start to question their relationship.
The relationships among the Heathers characters would be in flux when watching the musical fanfiction. Heather Chandler might see any deviation from her perceived 'right' portrayal as an attack on her leadership, and she could lash out at the others, thinking they had something to do with it. Heather Duke might try to use the fanfiction to her advantage, perhaps to undermine Heather Chandler if it shows her in a bad light, which would create more tension between them. Heather McNamara might be caught in the middle, trying to keep the peace while also dealing with her own feelings about how she is portrayed in the fanfiction. Veronica, who is on the outskirts of the Heathers' group, might use the fanfiction as a way to further distance herself from them if it shows their toxicity in a new light.
On the other hand, if a fanfiction creates some sort of false or negative scenario between them, it could cause some strain in their relationship. For instance, if a fanfiction wrongly implies that one character has betrayed another, there could be initial tension until they figure out it's just a fictional account.
Misa and Light's relationship might also change. In certain fanfics, Misa could be more assertive in her relationship with Light, which could lead to a shift in their power dynamic. Light, who is used to being in control, might have to adapt to this new Misa.
Characters' views on friendship could change. In the crossover fanfiction, they may see characters from different universes form strong bonds despite their differences. This could make them more accepting of those outside their own world or group in their original Naruto story. They may also change their views on power. If they see characters with very different power levels and types in the crossover, they might reevaluate what it means to be strong in their own context.
If they watch a horror series, they might cling to each other for comfort, strengthening their bond. For example, two previously distant characters could become closer as they share their fear during the scary parts of the show.
It depends on the character. For heroes like Metal Bat, he might get motivated by a fanfiction that shows him in heroic deeds that are even more epic than in the original. He could be inspired to train harder and be even more of a hero. But for some minor characters, they might just be indifferent. They might not have strong opinions because they are used to being in the background in the original, and the fanfiction doesn't really change their perception of their own unimportance much.
The characters would likely have a range of reactions. Those who are more serious - minded, like Uther, might be appalled at some of the more liberal interpretations of their world and actions. On the other hand, the younger and more open - minded characters, like Gwaine, could get a good laugh out of it and might even enjoy seeing new takes on their own personalities and the events they've been through.
If Merlin travels back in time, his relationship with Arthur would be very different. In the past, Arthur might not trust him at all at first. Merlin would have to prove himself all over again. With Gwen, he might be more aware of her future importance and try to protect her in a different way. And his relationship with the other knights could be completely new, as they might see him as a stranger with strange powers.