This isn't really a definite answer but I'd say it would help immensely if you had experience with Shakespeare plays. His plays contain many varied themes such as love, betrayal, jealousy, manipulation and racism. If you can portray characters with these themes and use them in your work correctly you will show good versatility.
It would be useful to have some knowledge about Shakespeare's plays, but it depends on what or who you are acting as.
I would say no and I am an actor at heart. Yes, I have done Shakespeare before but it didn't help me be a good actor. My acting professor taught us the essential things to be a good actor. I am not sure if you like to read but pick this book up. I think Amazon has it for like 2.99 on there and if not your local library would have it. Everything in this is what makes a good actor. "An Actor Prepares" by Constantin Stanislavsky. It is a good read also.
I don't see it as a prerequisite, however I'd think someone with true passion for any craft would want to take any opportunity they could to learn from the Greats.
Reminds me of this quote: "If I have seen further it is by standing on the shoulders of giants." It's attributed to Newton, but ironically he was borrowing even that from Bernard of Chartres who was centuries older still.
I'm an actor and I love Shakespeare. I think its because I am an actor at heart that both his stories and the beauty of the way he tells them, have such an impact on me. Most of my friends are actors - its a pretty incestuous community! Many of them love his work and lots don't - it's not a prerequisite to being an actor. Stanislavsky is the father of modern acting, but not the only one. Just do it! Do plays, classes, films, everything.
As an actor myself, I Can say you don't NEED it. But it is useful. It makes contemporary acting feel much easier.
The mere fact that the vocabulary and sentence structure is not average everyday English makes it a good exercise.