I don't think so. Throughout the story, we only know her as Mrs Shears. There's no indication of her full name at all. We can only guess what her first name might be, but the author didn't provide it in the story.
I'm quite sure it wasn't. Mrs. Shears is always called just that in the story. It's possible that the focus of the story was on other aspects such as the actions, relationships, or emotions of the characters rather than on providing every character's full name. Maybe the lack of her full name is a narrative choice to keep the story flowing and not get bogged down in too much detail about a character who may not be the central focus but still an important part of the plot.
No. When we read the story, we get to know her as Mrs Shears. There's no part of the story where her full name is brought up. It could be that the author wanted to keep her identity somewhat veiled or simply that her full name was not relevant to the story's main events. Maybe the story is more about the relationships and events that surround her without the need for a full name.
Barefoot Billy in the 'barefoot billy true story' might be a unique individual. He could be a person who rejected the norms of wearing shoes. Maybe he had a philosophical reason for it, like believing that being barefoot kept him more in touch with the real world. Or perhaps he grew up in a situation where shoes were a luxury he couldn't afford, and over time, it became a part of his identity.
Sure, Billy Bathgate is fiction. It falls within the realm of creative writing, where the author weaves a tale that may or may not have any resemblance to reality.