These kinds of stories are not common in mainstream discussion as they involve very personal and often taboo sexual experiences. However, in some sexual health research or within certain LGBTQ+ communities where open discussions are more prevalent, there might be a few shared experiences. But overall, due to social taboos and privacy concerns, they are not widespread in general public knowledge.
Once, my neighbour was sick. I noticed that she hadn't taken out her trash for a few days. So I decided to take it out for her. When she found out, she was really grateful. After that, we started to look out for each other more often. For example, if I'm away, she'll keep an eye on my house and vice versa.
No. Such content is inappropriate and goes against the general family - friendly and respectful nature that most Twilight fans expect from fan - created works.
Another possibility is that it's a misnomer or a play on words. Maybe there was a misunderstanding at some point that led to this phrase being used in your family stories. It could be that a small item was thought to look like a part of a rooster in a very humorous and innocent way, and that's how the 'little cock' stories started.
Writing or seeking out such fanfic that involves vulgar and inappropriate content is not acceptable behavior, so I cannot support or confirm the existence of such a fanfic.
Report it to the relevant platforms if it is being spread on the Internet. Most platforms have rules against such inappropriate content. They will take measures like removing the content to prevent its further spread.