A common theme could be adventure. Maybe Adam and Lute go on a journey together and face various challenges along the way. This could include battles against enemies or solving mysteries in a fantasy world.
There could be. Try searching on different fanfic websites. Some might be hidden gems. It really depends on how popular the Adam x Lute pairing is among the Hazbin Hotel fanfic community. If it's not a very common pairing, it might take more effort to find really good ones.
Common themes may include the exploration of their personalities. Since they are characters from Hazbin Hotel, their unique traits are likely to be delved into. Also, themes of loyalty and trust could be present, as their relationship develops. They might face challenges together that test their bond, which is also a frequently seen theme.
Another option is Archive of Our Own (AO3). It has a large collection of fanfiction. Just type in 'Adam x Lute' in the search bar and you'll get stories written by different authors. Some might be more focused on their friendship, while others could explore a romantic relationship between them.
A common theme could be self - discovery. Writers might explore how Adam Lute discovers more about himself, his powers (if he has any in the relevant context), or his place in the world.
Another option is to join relevant fan communities or forums dedicated to the work that Adam Lute is associated with. There, fans often share and recommend their favorite fanfictions about him.