Vegeta could be a great rival. He might see Goku's new position as God of Destruction as another thing Goku has achieved that he hasn't. This could fuel their competitive nature even more.
Goku does reach a state of immense power in the manga that rivals that of a God of Destruction. However, the specific title and all its implications might not be straightforwardly assigned to him. It's a complex development within the story.
Vegeta would be a great addition. His rivalry with Goku and his pride as a Saiyan would add an interesting dynamic. He might be initially jealous that Goku is getting to fight and train with the God of Destruction. But then he would also want to prove himself and join in the action. Whis could also be included. His knowledge and calm demeanor could help in situations where the power of Goku and the God of Destruction gets out of hand.
Yes, it can be. In fanfiction, Goku being born with God Ki can completely change the power dynamics. It could mean he starts off with a much higher level of power than in the original story, leading to different battles and interactions. For example, his training might take a different path as he already has this innate God Ki. He could potentially face stronger opponents earlier and form different alliances.
Goku becoming a God of Destruction could mean a new look for him. Maybe his Saiyan armor would transform into something more fitting for a god, with a more regal and powerful appearance. His power level would skyrocket, and he could start to have an influence on the balance of the universe. He'd also likely have to deal with other gods, which would lead to some interesting interactions.
One interesting concept could be Goku's transformation into a God of Destruction - like form. It might involve new powers and a different way of handling battles.
One idea could be Goku struggling to adapt to the role of God of Destruction. He's used to being a protector, so having to destroy planets as part of his new duty would create a great internal conflict.