One popular plot is the discovery of a common enemy that forces Superman and Homelander to team up, despite their differences. Another is a battle for supremacy in a world where both exist, with different factions taking sides. And then there's the plot where Superman tries to reform Homelander, seeing the potential for good in him.
Well, it depends on how the fanfiction is written. If it's a more 'grounded' version where Homelander's public image and mind - games play a big role, he could potentially gain an upper hand. But generally, Superman has more raw power. He can move planets, and his speed is almost limitless. Homelander is strong and has flight and heat vision too, but Superman just seems to be on a whole other level in terms of pure power.
One idea could be a battle in an alternate dimension where the laws of physics are different. This would force both Superman and Omni - Man to adapt their fighting styles. Another could be a scenario where they are forced to team up against a greater threat first, before their own conflict resurfaces.
It's hard to say for sure. Omni - Man has incredible strength and combat experience. However, Superman has near - invulnerability and his powers are powered by the sun. If the fight takes place on Earth, Superman has the home - field advantage with the sun constantly recharging him. But if it's a long - drawn - out battle, Omni - Man's tenacity could also be a factor. In my opinion, Superman's powers are more diverse and powerful, so he would likely come out on top.