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.
Kimiko is indeed present in the comics. Her character brings a lot of depth and intrigue to the plot. There are various storylines involving her that make the comics more engaging.
Kimiko's fate in comics can be quite complex. It might be that in one comic version she survives, while in another she doesn't. It all hinges on the narrative and the creative vision of the comic creators.
Yes, there might be. There could be fanfiction exploring their relationship in different ways, like a story where they overcome their differences and fall in love. However, it depends on the creativity of the fanfiction writers in that particular fandom.