Are content rating criteria applied to music videos?

In the US - Yes and No

The Music Industry itself has a voluntary Code & Ratings.

In 1985, the Recording Industry Association of America (RIAA), the National Parent-Teacher Association, and the Parents' Music Resource Center developed the black-and-white warning sticker now used voluntarily throughout the music industry. Still the least closely regulated of the major media industries, the label is voluntarily applied to albums containing coarse language or descriptions that would be inappropriate for children in the opinion of the recording label distributing the album. A small sticker is applied to the outside of physical packaging or on digital storefronts indicating simply “Explicit” or “Parental Advisory: Explicit Content.”

Music Videos though are covered in the US by the FCC for television performances

On the other hand, over the air radio broadcasts of music are among the most heavily regulated, with direct oversight by the FCC. All content must be within a certain range of appropriateness, with vulgar language and overly explicit content drawing fines and other penalties.

^Source

In the UK..YES

The new ratings system is a government-led initiative to prevent children’s exposure to inappropriate content. Among those on board include YouTube, Vevo, the British Board of Film Classification (BBFC), British record label trade body BPI and labels Sony, Universal and Warner Music. The program could be coming to the U.S. next.

All music videos in the U.K. must now be submitted to the BBFC — essentially the British equivalent of the Motion Picture Association of America — before they can be published online. They will then be vetted, assigned age-appropriate ratings before being submitted to YouTube and Vevo.

^Source