Are music videos under copyright law?

Yes. You may be subject to multiple agencies such as ASCAP and the RIAA if you play music videos without permission. Basically you would have to beg them for permission at which point they would send you the rates and then sue you for using the videos you've already paid for. I really wish I was making this up. Your only safe bet is to find unsigned artists who want you to share their videos. Professional videos just make you liable.