Maybe to some degree. As someone said, the shapes keep changing, but the art stays the same. All art is trying to find some way of making sense of the subjective realm of experience. Shakespeare just wasn't a story teller, he was interested in the mind and the 'soul'. I'd prefer more contemporary high art such as Mahler, or even Elgar. It has a nice 'enduring' quality despite being a 100 years old.
Nobody can possible know but if are to guess I'd say probably not. Shakespeare was from near Manchester which is really big on rock bands like Oasis, Stone Roses and The Smiths. I think he would have been into those bands.
As for Dickens. He wrote Oliver Twist who was a bit of a rebel. So I think he would have been into punk rock music.