One possible plotline could be that Broly, in his rage, accidentally provokes Beerus. Broly, not knowing who Beerus is, attacks him. Beerus, initially amused, decides to teach Broly a lesson but soon realizes that Broly's power is growing at an alarming rate. This leads to an intense battle where Broly pushes Beerus to use his full power. Another plotline could be that Beerus hears about Broly's power and decides to test him. He travels to Broly's location and challenges him. Broly, seeing Beerus as a threat to his friends or his planet, fights back with all his might.
Beerus would probably win. He has the status of a God of Destruction, which means he has a certain level of power and abilities that are unique. His Hakai ability can erase things from existence. Broly is strong, no doubt, but he is still a mortal (in a sense). Beerus has been shown to be far above most mortals in terms of combat ability and raw power. In a fanfic, unless the writer completely changes the power dynamics, Beerus has the edge.
I'm not entirely sure specifically what a 'beerus raps beerus fanfic' is about without more context. It could potentially be a fan - created story where the character Beerus is rapping. Maybe it involves Beerus expressing his thoughts, his powers, or his experiences in the form of rap within the context of a fan - made fictional narrative.
You could try looking on fan - fiction websites like Archive of Our Own or FanFiction.net. They usually have a wide variety of fan - created works, and there's a good chance you might find a 'beerus raps beerus fanfic' there.
Beerus would likely be more powerful. He is a god of destruction in his own universe with incredible cosmic powers. Naruto is extremely powerful in his world, but his power is mostly based on chakra and ninjutsu, which is on a much smaller scale compared to Beerus' ability to destroy planets with a flick of his finger.
With King Kai, Beerus as a girl could have a different kind of relationship. She might be more patient with him. King Kai might be a bit more nervous around her at first, but then they could develop a more comical and friendly rapport, with Beerus teasing him in a more light - hearted way.