Is this expression equivalent to "it makes me happy"? Does it sound ok or maybe too old style? Would I ever hear a young person saying it? Is this expression used or common?
"X macht mir Freude" corresponds more closely to "I like doing X". "X makes me happy" would be "X macht mich froh/glücklich".
And to express "I like...", "Spaß machen" is somewhat more usual than "Freude machen", although it's still perfectly understandable.
Es macht mir Freude is grammatically correct, but sounds a little bit strange in everyday verbal communication.
I'd use Es freut mich.