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.
Charlie is often paired with Lute. Since she is the owner of the Hazbin Hotel and very passionate about redemption, she might see Lute's music as a tool for helping the sinners, and they could work together on different projects.
There is a recommended novel that meets your requirements, and that is Mr. Wei's Miss Pipa. The heroine of this novel was the lute girl, and the lute played a very important role in the story. I hope you will like this novel.
In a fanfic of Adam and Eric from 'Sex Education', it could also delve into their family backgrounds and how that impacts their relationship. Maybe Adam's difficult relationship with his father makes him more reliant on Eric's positive family environment for emotional support. And Eric, in turn, might learn about standing up for himself more from Adam's experiences of dealing with bullies.