To be honest, I only half get what you are trying to get at, but I'll see if I can get what you think I've got while the going's good! Glad I got that out of my system
Shakespeare's relevance today would be moreso than a common 'joe' may realize. But my opinions to your questions are: yes, slightly, and no. They are highly accessible, but not presented in a fashion to promote appreciation and understanding in a fashion that would be similar to that of Shakespeare's contemporary theater-goers.
Many modern plays can be traced to Shakespeare. Such as most recognizably "Westside Story." The movie (and broadway production) have had enormous success, regardless of age demographics. Many other stories have their plots loosely based on Shakespeare (the chick flick "She's the Man").
I'm also pretty sure that music artists and rappers use Shakespeare's forms for creating lyrics. But I'm very likely wrong. Second opinion please!
Shakespeare, as we know, wrote in iambic pentameter. In looking at his early pieces, we see the standard and straight poetry. This was the common practice at the time for theatre as we can see by his predecessors/contemporaries. In his later works, he had not forgone it, but modified it by moving an iamb to a different line, thereby creating a more natural speech pattern. He also added "accidentals" which further helped flow. This of course can trace to the lack of poetry in many modern shows. A far cry from Greek plays.
The relevance to modern youth in that regard, is negligible, but if you find it useful...
If you have the chance, I would advise looking at "The Taming of the Shrew" by the San Francisco Conservatory Theater.
Anyways, his plays have had an impact on literature and performance arts. As we can see from the aforesaid production, though the words remain the same, the meaning can be conveyed through actions as well as words. As long as the meaning and subtext can be said, the story is as deep as when written. Shakespeare wrote on subjects that often had a very humanistic quality, if hidden somewhat. (Compare the French reaction to Tartuffe, to that of Scaramouche [quote from Molière's preface to Tartuffe:
Eight days after being banned, a play called 'Scaramouche the Hermit' was performed before the court; and the king, on his way out, said to this great prince: "I should really like to know why the persons who make so much noise about Molière's comedy do not say a word about Scaramouche." To which the prince replied, "It is because the comedy of Scaramouche makes fun of Heaven and of religion, which these gentlemen do not care about at all, but that of Molière makes fun of them, and that is what they cannot bear."])
The meanings that Shakespeare would put out, would be obvious enough for the masses to understand, but not so overt as to lose patronage of the royals. These political statements can A) give us insight to the politics of the time B) state and reinforce profound truths we know about the human condition (Romeo and Juliet's theme that love transcends all barriers, set forth the modern definition of 'love').
The way that Shakespeare is taught here in America, in literature classes at least, is shameful. Often the word plays are left out, or the meaning is somewhat lessened. Many large film productions I personally dislike. Shakespeare had a large amount of commoners attending his shows. (Btw, the pronunciation at this some was still not fully Elisibethian. If I remember, it would have been similar to pirate-ese. But not Chaucer status. Knight is "n-I-t" but would have been "k-n-EE-cht") So, it would have been similar to the Larry the Cableguy of his generation. IMO.
If it was taught how I feel it should be, euphemisms and nuances included, the general populace would find Shakespeare as moving as he was in the 1600's. If I remember, the naysayers said he "incited the crowds". But you will need to double check that.
And his lead actor was a master swordsman. It would have been awesome. He would have had proper medieval technique, making for strong, physical scenes. And according to those around him, he was very charismatic. While speaking in playful or haunting poetry. And if he was trying to make everyone understand what was going on (same as if you mute the telly and try to follow the story), it would be very compelling.
Action, thriller, comedy, romance, mystery, ect. He made them all. By no way is he done. His name will live on. And even when we forget many of the accomplishments of that generation, we will still have Shakespeare. He is just that important. And he should be! But how it is taught in American public schools is....very poor form.
So he is accessible and relevant so long as we let it be. If we read it as just words(without nuance and sub-text), it become the "meh" many see it as. But if we try to say what the Bard wanted to say, it is. So. Much. More.
Just my two cents. Hope it helped. Even if it was/n't what you were looking for. I could try again if you want, so long as you are more specific.
just my BS-ing attempt