I think it depends on various factors. Goku's determination and growth are remarkable, but Beerus has an overwhelming level of power and authority that might be difficult to overcome.
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.
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.
Generally, Goku still holds the edge over Vegeta. Their power levels fluctuate, but Goku often achieves higher peaks of strength and unlocks more powerful forms first.
His character would develop a more strategic side. Beerus is a very strategic fighter, and Goku, being raised by him, would learn to think more before acting in battles. He would also become more aware of the power hierarchy in the universe at a younger age, which could make him more cautious or more ambitious, depending on how you look at it. Additionally, his Saiyan pride might be tempered by Beerus' teachings, making him more of a team player in some ways.