To get something started, I'll add my perspective as a classicist and Shakespeare enthusiast.

The thing about rhetorical devices and the like is that you don't need to notice them for them to work. In fact, it's all the better if you don't sit there and go "Aha! Anaphora!" It doesn't matter if you miss the device if you get the effect. Let's take an example from perhaps my favorite comedy, Twelfth Night:

Feste: Good madonna, why mourn'st thou?

Olivia: Good fool, for my brother's death.

Feste: I think his soul is in hell, madonna.

Olivia: I know his soul is in heaven, fool.

We often see Shakespeare's characters in battles of wits, and it's not always a fair match, particularly when the fool archetype (e.g. Feste) is involved. From this brief exchange, we see that's not the case. Feste and Olivia are equally well armed. Assuming the lines are delivered well, the audience can understand that fairly easily.

When we take these lines into the classroom, we see it as a classic example of isocolon: the repetition of phrases with the same structure. This rhetorical device provides balance and emphasis to one's writing. It's simple and easy to spot, yet many of the most remembered phrases of literature and speeches contain it: Caesar's "Veni, vidi, vici," Lincoln's "We cannot dedicate – we cannot consecrate – we cannot hallow..." and "government of the people, by the people, for the people..." You don't need to know that Lincoln was using isocolon to get that something's really important about the connection between a government and its people.

Going back to your example, the prologue of Romeo and Juliet, we again see isocolon.

Two households, both alike in dignity,

In fair Verona, where we lay our scene,

From ancient grudge break to new mutiny,

Where civil blood makes civil hands unclean.

In this last line, it's obvious that Will wants us to know these are civil parties. He could have said "very civil," or used "civil" again in the next line, or even in the next three lines. But by using an isocolon's parallel structures and rhythm, he really drives home the point. These "rhetorical flourishes" are really just the way good authors put their words together.

It's kind of like the special effects and cinematography of a film. There's any number of ways to film someone walking through a door, and how you do so tremendously impacts the viewer's perception of the action. Do you go with a dramatic low angle shot with the door taking up the full frame showing the hero walking slowly through? Do you do a close up of the door handle turning and follow the actors hand with its identifying ring for a bit of suspense? Film sound designers pull all kinds of noises to get the exact one they want- who in the audience hears Chewbacca and thinks "Golly! Those animal sounds Ben Burtt together sound great!" In the end, it doesn't matter exactly how you do it as long as it's got the right effect and look. But if I'm in film class, these are the things I want to know. What is it about Hitchcock's angles make his films work? How do I mix sounds I don't want to make one that I do? When we study Shakespeare academically, we pick apart all these devices for the same reasons: to study what makes him unique, to improve our own writing, to compare him to other playwrights. When we're in the audience, all that matters is the effect. And of course, sometimes that effect will get lost on people. But this true in any work. I don't catch all the pop culture references in even my favorite sitcoms, and in the same way we can't expect Shakespeare's audiences to have caught each hidden meaning or literary allusion.