No. It's not a common type of story at all. Most traditional stories don't involve such a grotesque and unusual image of eels going down someone's throat.
Hardly. Common stories are based on human experiences, emotions, and social interactions. This kind of story about a 'cock disappearing down her throat' is so far from the norm. It doesn't fit into the usual categories of stories like coming - of - age, horror in the traditional sense, or even science - fiction. It seems more like a very idiosyncratic and not very common concept for a story.
It could be a very strange and perhaps disturbing or surreal image in a story. Maybe it represents a feeling of being invaded or overwhelmed, like the eels are an unwanted presence forcing their way into the girl.
I'm not entirely sure without more context. It could potentially be a very strange or disturbing story about an animal (a rooster perhaps) in some sort of fictional or very odd situation. But it's also possible it could be misinterpreted as something inappropriate if 'cock' is taken in a vulgar sense.
We could use 'tales of positive human interactions' to replace it. This kind of phrase is inclusive and focuses on positive aspects of human relationships rather than inappropriate content.
In the metaphorical sense, inexperience. If it's your first attempt at something like a new job or a new relationship (your 'first story'), you may not have the skills or knowledge yet, and this can cause things to go wrong and seem like they are 'falling down'. For example, in a new job, not knowing the proper procedures can lead to mistakes that can start a downward trend in your performance.