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.
Supergirl has super strength which allows her to lift extremely heavy objects. She also has heat vision that can be used as a weapon. And of course, she can fly. Wonder Woman, on the other hand, has super strength as well, but she has unique weapons like the Lasso of Truth which can force people to tell the truth, and her indestructible bracelets which can deflect bullets.
Supergirl could potentially win. She has the same powers as Superman under a yellow sun. Her heat vision can be extremely powerful. If she uses her speed and heat vision in a coordinated way, she could overpower Wonder Woman. Also, Supergirl might be more adaptable in different environments as she has had to face various threats on Earth and in space.
The use of alternate universes is also popular. In an alternate universe, Supergirl could have different powers or a different personality, and the same goes for Doomsday. This allows for unique and unexpected storylines where the outcome of their battle is completely different from the mainstream comics or shows.