Do today's music, fashion, and TV/movies challenge and change the social norms in the same way that music in the 1960s influenced people to change...

Every generation defines itself in relation to the generation that came immediately prior. The young people of the 1960s were children of parents who survived the Great Depression in the U.S. and they sought to create lives that contrasted with the lives of their parents. The older generation had gotten young men (mostly) engaged in a war that seemed to be a fruitless waste of human life and some of that older generation resisted the changes brought along by the Civil Rights Movement. Fashions of that period of time, the music, and other forms of popular culture reflect these conditions and more. For example, young men began wearing their hair long as a contrast to the clean cut look of both the 1950s and the Armed Services. Protest songs reflected social conditions during the time, as did rock and roll, particularly as rock became more avant garde. Television and movies were more conservative but they still responded to the youth movements, whether through expressing support, bewilderment, or downright dislike.

At the same time, not only did popular culture respond to youth culture, but it also drove youth culture. Music brought new people into awareness of social conditions, which inspired some to join the counter culture. Television shows such as Rowen and Martin's Laugh-in made groovy partying into a good thing. 

The same is true of today's young people and popular culture. Our high-tech age makes certain kinds of movies (think special effects), music, fashions (wearable tech), and so forth possible, so that is one way that cultural production influences people's lives. Imagine what life would be like without social media, which has profoundly changed the lives of many young people, particularly in contrast to the lives of young people in a time where the height of tech was having a stereo phonograph. Another influence is the fact that the conditions that young people face currently are very different from those faced in the 1960s. Today's economy is struggling a lot more than it did in the post–World War II years and today's young people not only have trouble finding a job, but they also may have significant debt for a college education. A significant line of thinking in the 1960s was utopic; this is where communes came into play.

Of course, we have about forty years of intervening history between today and the height of the 1960s, so we have the value of hindsight in considering the culture of the 1960s. In about 40 years, we should be able to provide many more examples; in the meantime, the answer to your question is yes, popular culture influences each generation and also that each generation influences popular culture.