His power would develop more in line with Saiyan traditions. He'd probably have a stronger base power due to the intense training environment on Planet Vegeta. There would be more emphasis on raw power and combat skills from a young age.
The power level would be insanely high. Naruto already has access to a huge amount of chakra, especially in his later forms like Six Paths Sage Mode. Vegeta, on the other hand, has his Super Saiyan forms which multiply his power exponentially. When fused, they would likely have a power level that could easily rival some of the most powerful beings in both universes.
In a fanfic, Naruto might power up by having a spiritual awakening. He could connect with the spirits of past Hokages in a deeper way, gaining their knowledge and some of their power. This could enhance his existing jutsu and give him new abilities. Vegeta could power up if he were to face a near - death experience against Naruto. This could trigger a hidden potential within him, like a new type of Super Saiyan form that has unique abilities. He could also fuse with another Saiyan (if there were another Saiyan available in the fanfic's scenario) to create an even more powerful being.
Bulma could be involved. Since she has a relationship with Vegeta and is friends with Goku, she might try to mediate between them when Goku snaps at Vegeta. She could be the voice of reason, trying to calm both of them down and make them see each other's point of view.
It often starts with a shock or surprise at the mpreg situation. Then, they gradually learn to rely on each other more. For example, Vegeta might start to show a more caring side that he usually hides, and Goku might become more responsible. They'll have to communicate more, sharing their fears and hopes about the future with the baby.
Yes, in the vast world of fanfiction, there are all kinds of creative and out - of - the - box stories. Some fans like to explore unusual relationships and scenarios, so it's possible that there is such a fanfiction out there. However, it goes against the original canon of the Dragon Ball series in many ways.
Well, if they were brothers in a fanfic, it could change how they interact with the other characters. Maybe they'd have a deeper understanding and fewer power struggles between them. Also, their family background and upbringing could be explored in new and interesting ways.
It's a bit hard to say for sure. Both Vegeta and Goku have their moments of strength. Goku has some unique powers and techniques that give him an edge at times, but Vegeta is no slouch either and constantly works to catch up and surpass Goku.