Yes, there are likely many fanfics about Karen Wheeler from Stranger Things that are family - friendly or focus on other aspects like her relationships in a non - sexual way. You can try searching on fanfic platforms and filtering by appropriate tags.
One possible storyline could be Karen Wheeler's discovery of the strange happenings in Hawkins from a mother's perspective. She might be worried about her kids and try to protect them in her own way, without any sexual undertones. Another could be her getting involved in community efforts to deal with the aftermath of the Upside - Down incidents.
Billie could be developed by giving her some new abilities or skills. For instance, she could have an uncanny sense for detecting when the Upside Down is about to seep into the real world. Mrs. could be developed by showing her as a mentor figure. She could start teaching the younger characters in Hawkins some valuable life lessons that also tie into dealing with the supernatural threats.
One plot idea could be that Billie and Mrs. form an unlikely alliance to solve a mystery in Hawkins that the main group has overlooked. Maybe they discover a hidden connection between the Upside Down and a local family's history.
Yes, there are. The Stranger Things comics offer additional adventures and backstories that complement the TV series. They provide more depth to the Stranger Things universe.
One common theme could be the relationship dynamics between Mrs Wheeler and Billy. It might explore how their personalities interact and develop over the course of the story. Another theme could be the setting, like the GH (presumably a certain location) which could play a role in shaping their encounters.
I'm sorry, I'm not very familiar with specific 'Mrs Wheeler and Billy Rough Lemon' fanfiction. You could try searching on popular fanfiction websites like Archive of Our Own or FanFiction.net. There might be a variety of stories there related to them.