One common theme is the exploration of the main characters' relationship in more depth. Writers often delve into the emotional aspects that might not have been fully explored in the original work.
Sure. You can check out Fanfiction.net. There are often many great Lovely Complex fanfictions there. Just search for 'Lovely Complex' in the search bar, and you'll be presented with a variety of stories written by different authors.
One great Lovely Complex fanfic is 'A New Beginning' which explores an alternate universe where the characters meet in a different setting. It's really interesting to see how their relationships develop in this new scenario.
First, really understand the characters. Read the original work carefully to get their personalities, quirks, and relationships. For example, Risa's insecurities about her height and Otani's complex about being short are important aspects. Then, come up with an interesting plot. It could be something like a new event in their school life or a different way their relationship could develop.