Most employers do at least a basic check of your work history and references. You would probably be caught and lose the offer. At best you might get away with a lie by omission when it comes to your age. If you can avoid telling them what year you graduated, they may assume you are younger than you are and be a lot more forgiving of your lack of work history. Don’t outright lie though, they will see your true age when they take your ID after hiring you.
As for volunteering, choose anything you like. Though I would avoid religious or political organizations. Any work that shows you are able to make a commitment and handle responsibility will look good on a resume.