Definitely not. Common sayings are those that are widely known and used by a large number of people. 'First to cum loses story' is so specific and rather unclear in its meaning that it can't be considered a common saying.
A 'a mouthful short story' might refer to a short story that contains a lot of information, perhaps with rich characters, intricate settings, and a detailed plot. For example, it could be a story that tries to pack a lot of action or emotional depth into a few pages, making it feel like there's a lot to digest, hence 'a mouthful'.