Yes, there are. The Naruto fanfiction community is vast and creative. Some writers like to explore the idea of Naruto having Wood Style, often by creating unique storylines that might involve him learning it in a different way than the original series, perhaps through some new form of training or a special encounter.
The Legend of Uzumaki Hiroto was a derivative work of Naruto, which told the story of Uzumaki Hiroto's growth and the stories of other characters. Uzumaki Boto Legend was different from the original Naruto Naruto Shimbun Legend, so it was considered an independent manga.
One interesting fanfiction could be 'Naruto: The Wood Style Prodigy'. In this, Naruto discovers he has an affinity for Wood Style early on and has to hide it from the village at first. Another is 'Wood Style Naruto: A New Path' where he learns Wood Style from an ancient scroll and uses it to revolutionize the way ninjas fight in the Hidden Leaf. There's also 'Naruto's Wood Style Adventure' which focuses on Naruto using Wood Style to explore uncharted regions of the ninja world.
It depends on the specific plot of the fanfic. Sometimes he could be on a secret mission, in a hidden village, or facing a new challenge in an unknown location.
One idea could be a character from Fairy Tail who somehow gains the ability to use Naruto's Wood Style. For example, Natsu could discover a magical item during one of his adventures that gives him this power. He might initially struggle to control it as it's a completely different type of magic/ability compared to his Fire Dragon Slayer magic. But as he practices, he could start using it in unique ways, like creating wooden constructs to fight enemies or using the wood to trap opponents in a forest - like environment.
Well, he could start a family with Hinata and have lots of kids, passing on his Uzumaki bloodline. Then, he could encourage his friends who have some Uzumaki connection, like Karin, to also have families and pass on their Uzumaki genes. He could also set up a school or training center in the Uzumaki land, teaching the Uzumaki ways of life and ninjutsu, which would attract orphans or those who want to be part of the Uzumaki clan.
The Wood Style would introduce a whole new element of combat. In Fairy Tail, most of the magic is elemental - based like fire, water, etc. or celestial - based. Wood Style could provide a more strategic and defensive approach. For example, a character using Wood Style could quickly create barriers or traps made of wood during battles. This would force the opponents in Fairy Tail to adapt their fighting styles.
They are not appropriate. The Naruto series is known for its action, adventure, and positive themes. Lemon fanfics that may contain sexual or explicit content are not in harmony with the family - friendly nature of the original Naruto work. It's better to focus on fanfics that respect the original themes and values of the series.
A really interesting plot could be Naruto traveling back in time to the era before he was born. He could meet his parents and other important figures in the village's history. This would give him a chance to understand the roots of the village better and also learn some new jutsu or strategies that he could bring back to his own time to make the Hidden Leaf even stronger as a Hokage.
Naruto could also express his love in a more playful way. He might tease the reader a lot but in a loving manner. And he'd bring her little gifts, like a special kunai he found on one of his adventures or some flowers from the outskirts of the village. He'd want to make the reader laugh and feel special all the time.