Naruto's development would be a very interesting mix. In Kumo, he'd be exposed to different types of chakra natures and fighting styles. His Rasengan, for instance, could be modified with Kumo's lightning chakra to create a more powerful and unique attack. Minato's influence would ensure that Naruto has a solid foundation in space - time ninjutsu. Also, being in Kumo, Naruto might learn to control his Nine - Tails chakra in a different way, maybe with the help of Kumo's unique methods of chakra control. This would make him an even more formidable ninja compared to his original development in Konoha.
In this kind of fanfiction, Naruto's life would be drastically different. He might be trained in Kumo's unique ninja arts. Minato being alive could mean he has some sort of connection or influence on Naruto from afar. Maybe Naruto would grow up with a different set of values influenced by Kumo's culture, while still having his father's legacy in the back of his mind.
Naruto's personality would be different. He'd be less lonely - driven, as he'd have Minato's love. He might be more focused on being a great ninja like his father from the start.
Naruto's character would develop in a more balanced way. Minato was calm and intelligent. So Naruto would learn those traits from him. He wouldn't have the same insecurities as in the original story. He might be more of a natural leader from the start, as Minato was also a respected leader. And he would have a better understanding of the village's politics and history because Minato would be there to educate him.
His chakra control would likely be extraordinary. Kyuubi is a huge source of chakra, so Naruto would have had early exposure to vast amounts of it. This could lead to him being able to perform more advanced jutsu at a younger age.
Naruto's character would develop a different sense of loyalty. Instead of being loyal to Konoha, he'd be loyal to Kumo. His goals would change too. Maybe instead of aiming to be Hokage, he'd aim to be one of the top - ranking ninjas in Kumo. His social skills could also be different. Kumo has a different social environment, so he might be more rough around the edges in his interactions.
His personality might be more rough around the edges. Kumo has a tougher environment, so he'd likely be more assertive.
His character would be more confident. Growing up with parents, he'd have that support system. He wouldn't be as desperate for attention as he was in the original. Instead of seeking validation through pranks, he'd get it from his family. His ninja skills would also develop more smoothly as he'd have Minato's direct training from an early age.
His character might become more hardened. Kumo is known for its tough ninjas, and living there would expose Naruto to a more cut - throat environment. He may lose some of his naivete and become more calculating in his actions.
His character could become more adaptable. In Kumo, he'd be exposed to different values and ways of life. This would make him more open - minded. For instance, he might start to view the concept of power in a new light, as Kumo has a different emphasis on strength compared to Konoha.
His physical abilities would likely improve a great deal. Black Widow is a master of martial arts, so Naruto could become even more formidable in close - quarter combat. His agility and strength could be enhanced beyond what he achieved in the ninja world just through her training methods.