No, it's not common. In most schools, there are dress codes that require students to wear proper footwear for safety and hygiene reasons. In the story, if she is barefoot, it's likely an exception due to the reasons mentioned before like losing shoes or having an injury.
It's interesting because it's different from the common school scenarios we usually think of. Most of the time, we picture students in proper shoes and uniforms, so the idea of being barefoot adds an element of novelty.
She probably felt embarrassed. Walking around without shoes in school is not normal and she might be worried about what others would think of her.
It could be that someone played a prank on her and hid her shoes. Some students like to do naughty things like that. There's also a possibility that her shoes were old and the sole got detached suddenly, making it impossible for her to wear them.
There's no way to tell for sure who the barefoot girl is without more of the story. She could be a fictional character created to teach a lesson, like a poor girl from a small village who values the shoes she has and is distraught when she loses them.
It's not very common. One reason could be that she might have forgotten her shoes in the gym locker. Another possibility is that she has some discomfort in her shoes and prefers to go barefoot for a while. Maybe she's in a hurry and didn't have time to put her shoes back on properly.
Maybe she lost her shoes. It could be that she was playing too hard during recess and her shoes got misplaced. So she had to be barefoot in school until she found them or got a new pair.
Perhaps she doesn't like the idea of being teased because of being ticklish, so she denies it. Ticklishness can make a person seem more vulnerable, and she may not want to give others the chance to use that against her.
No, I don't think it is a common story. I've not come across it frequently in mainstream literature or general storytelling.
There's no way to know for sure without more context. She could be anyone. It could be a fictional character created for a story, or it could be based on a real - life situation where an unnamed girl was in such a predicament.
Maybe she lost her shoes. It could be that they got stolen or she accidentally left them somewhere.