http://en.wikipedia.org/wiki/Billy_Elliot_the_Musical
it briefly talks about the synopsis here. I haven't seen the film or play, but I know it was rated R, and I have heard the play stays pretty true to the film. There are riots/fights with police, boys dressing up as girls and some other things people may find difficult to explain to children, I've heard it's wonderful for adults with topical themes that many children may not understand. There is a lot of profanity, many of it spoken by the child actors. Some critics say it is family friendly, others say it isn't.
As for American Idiot, I have heard it compared to Rent.. it's basically about a group of friends who seperate, some do the rebellious anarchy thing for a while, another goes off to war, then they eventually get back to each other, obviously with the Green Day songs infused within. I've read if you aren't a Green Day fan, you would probably enjoy something else.
American Idiot may be inappropriate for children ages 12 and under. It contains adult situations, simulated drug use, strong language, strobe lights, theatrical haze effects, and herbal cigarettes. From here:
http://www.telecharge.com/BehindTheCurtain.aspx?prodid=7944