Sure is! Batman has been a staple of DC Comics for a long time. His stories and adventures are part of the DC universe. DC has built a rich and complex world around him.
It's hard to say for sure if it's rare. You'd need to look at things like how common it is on the market, if it's a first printing, and if it has any unique characteristics that make it stand out. Without that info, it's tough to give a definite answer.
Well, it depends. If you like the more traditional and gritty side of Batman, then it could be a good start. It might introduce you to his rogues' gallery, his relationship with Alfred, and his detective skills in a way that's engaging for new fans. However, some might find other Batman stories more beginner - friendly depending on their taste.
300 isn't a DC comic. It has its own independent creation and doesn't fall under the DC Comics universe. The story and characters are distinct from DC's offerings.
There are a ton of Batman DC comics. Different publishers, storylines, and reprints make it impossible to say exactly how many. But you can bet it's a huge number!