They are popular because fans love seeing the father - son relationship between Naruto and Boruto explored. Naruto is a beloved character, and seeing him show affection to his son in fanfictions is heart - warming.
There's a fanfiction titled 'Unexpected Hugs'. In it, Boruto gets into a bit of trouble during a mission and Naruto, who comes to his rescue, hugs him to comfort and scold him at the same time. It's a really heart - warming story. Also, 'The Healing Hug' is a fanfic where Boruto is injured and Naruto's hug gives him the strength to recover.
There are. You can also try Archive of Our Own (AO3). In some fanfics, the hug between Sakura and Naruto could be a turning point in their relationship. It could be a moment of realization for either of them about their feelings towards the other. Maybe Sakura hugs Naruto in a moment of vulnerability, and this act makes Naruto see her in a new light. It's really interesting to explore the different ways authors interpret this relationship through their writing.
Physically, Naruto can use his Rasengan or other powerful jutsus to fend off enemies attacking Boruto. His combat skills are top - notch, and he won't hesitate to engage in battle to keep Boruto safe.
In the 'boruto kills naruto fanfiction', a motive could be a power struggle. Boruto might believe that Naruto's power is holding back the village's growth or his own potential. He could think that by eliminating Naruto, he can bring about a new era of power in the Hidden Leaf. Also, it could be out of desperation. If Boruto is in a situation where he believes that killing Naruto is the only way to save someone he loves, like his mother or a friend, he might take such a drastic step.
Maybe it's the high expectations that Naruto has for Boruto. Naruto is a hero, and he might unknowingly put a lot of pressure on Boruto to follow in his footsteps. Boruto could be scared of not being able to meet those expectations.
They could start by having an honest conversation. Naruto could sit down with Boruto and just listen to his fears without judgment. This would be a first step in building a better relationship.