There's not enough information to know for sure who Bernie is. Bernie could be a person, a pet, or even a fictional character created just for this bedtime story.
I'm not sure. 'Real' is a relative term. Bernie might be a blend of imagination and real-life influences, making it sort of real in a way. But without more details, it's hard to give a definite answer.
In Pulp Fiction, if there is a Bernie, he is not one of the most prominent characters like Vincent Vega, Jules Winnfield, or Mia Wallace. It could be a character that was added for some extra flavor or local color in the complex world that Quentin Tarantino created. Without more context, it's difficult to say exactly who this Bernie is.
It's hard to say for sure. Sometimes stories draw elements from reality but are still mostly made up. Without specific information, it's tough to determine if Bernie is based on a true story.