We will haste us.

Probably what they should have said FIRST. Probably didn’t need to speechify about a bunch of kingly things. Probably should have said, “Yes, sir. We’ll go to England as you asked. Thank you very much. We will haste us, goodbye.”
But. . .it takes all kinds. And it can take time for people to get a hint. Like, the people who just won’t hang up or who won’t leave or who won’t let you leave. Maybe Rosencrantz is one of THOSE.

Never alone did the king sigh, but with a general groan.

This is why I’m glad to not be a king.
Being able to sigh on your own
just whenever you feel like it
without really bothering anyone;
Well this is not to be underestimated
I imagine that some might enjoy it – enjoy the constant attention and responsiveness of people around them. And generally, those are the kind of people who want to be king. Which makes me think about Utah Phillips and that little speech he gives about anyone who wants to be president being just the kind of president we don’t need.
He says something like the best presidents have been the Do Nothing Presidents.
Which is perhaps kind of a flaw in our democracy. Because giving rulership to a guy who wants it the most selects for a certain kind of leader. Just handing down leadership to anyone who happens to be next in line allows for some diversity in leadership styles. At least a little bit – because, of course, those that want to be king will find a way. That’s why we get such stories as are in Richard 3 or Henry the 4th or Richard the 2nd and so on.
And this oft quoted line serves them all.

Which when it falls, each small annexment, petty consequence, attends the boisterous ruin.

Uh? Rosencrantz? I’m not sure you’re making this situation any better.
You’re going from the king’s very personal death, to mass destruction and ending with a boisterous ruin.
I mean. . .weren’t you guys just talking about sending Hamlet to England?
How in the world did you end up at boisterous ruin? Well, shortly – you end up at general groan. . .which is slightly less apocalyptic than the death and boisterous ruin.

This speech is often cut, for many many good reasons – chief among them the way it adds absolutely nothing to the plot of the scene. It does, however, add an odd little something about Rosencrantz’s character. Who is this guy, when talking about taking this guy’s stepson out of the way, goes on an epic dystopian riff about the death of the man he’s talking to?

The sidekick doth protest too much, methinks. Is he trying to convince himself that taking Hamlet to England, (and possibly he knows that England is code for taking him to his death) is the right thing to do? The worse the circumstances are the more justified he is in doing it. It’s got to be apocalyptic, I guess.

Or ‘tis a massy wheel Fixed on the summit of the highest mount, To whose huge spokes ten thousand lesser things Are mortised and adjoined;

There’s the wheel. This one’s massy. Or mossy? It’s massy.
It is not a weal. It is a wheel.
Surely that will be heard and clear! (Surely not.)
This is a bizarre metaphor.
A giant wheel? At the top of a mountain?
That’s got a whole bunch of little things attached to it? Huh?
It’s not like – say, a cart?
Or – some actual wheeled thing?
No – I picture something like a bicycle wheel with lots of stuff stuck in between the spokes, playing cards, trinkets, tassels and decorative plastic flowers – and this weird wheel is hanging out at the top of a mountain –
Maybe little strings hang down from it and stuff moves when the wheel turns, like puppets attached to their controls.
What the hell is Rosencrantz going on about?

The cess of majesty Dies not alone, but like a gulf doth draw What’s near it with it;

I think, Rosencrantz, it’s probably not good etiquette to mention the king’s death to the king, even philosophically. No one’s particularly keen to be reminded he is going to die, but especially kings and (you can’t know this) but especially kings who have committed regicide. So you’re reminding him, Rosencrantz, of not only his OWN death but the death of the previous king, his brother.
It’s just not good form.

But much more That spirit upon whose weal depends and rests The lives of many.

Weal is a funny word – because, when spoken as this is meant to be, it sounds just like “wheel” and in meaning, yes, weal, as in well being might actually make more sense – but as a metaphor, the many lives might just as easily roll along on the spirit of a wheel. like a train or a car.
And in fact, when this speech gets going, it goes EXACTLY there. There is a weal, then a wheel – like Rosencrantz got to the wheel metaphor from using the weal one. Now all we’d need to get a real confusing pun-tastic speech going is to get someone with a little speech impediment to talk about something real. You know, someone who can’t really say his Rs, so he talks about shows that are weally weally good.

The single and peculiar life is bound With all the strength and armor of the mind To keep itself from noyance;

My particular single and peculiar life
Does not seem quite free from noyance

Is anyone’s life free from it?

A noyance – free life would be so swaddled in cotton, there’d be no edges – and at a certain point, the cotton itself would become a noyance.

I had a book of Andy Warhol quotations and for some reason a line of his about rich people and problems stuck with me. It was something like, “Everyone has problems. Even rich people – even if it’s just that the toilet doesn’t work.”
Which frankly would be a bigger noyance for the poor as it will take either more time or a larger percentage of an income to fix it. But it is a noyance in any case.

But – any single, peculiar life does STRIVE to keep itself free from noyance. Even the simplest life forms move toward the nice things away from the not nice. We have movement and brains to help us do just that.

How can that be, when you have the voice of the King himself for your succession in Denmark?

And here is where Rosencrantz signs his own death warrant. He might have been able to pull Hamlet’s allegiance back at this point with a different answer.
He’s been pushing at Hamlet to tell him what’s wrong pretty every time he sees him and here Hamlet finally tells him something trued and Rosencrantz shuts it down.
If he’d said something like, “Yes, that is upsetting.”
Or even just did some straight up reflective listening like, “You lack advancement.” Hamlet might have begun to trust him – but by essentially taking Claudius point of view, he condemns himself.
It is here where Hamlet quits playing their game, gets out the pipes and starts turning the tables on them.
Smooth move, Rosencrantz.

You do surely bar the door upon your own liberty if you deny your griefs to your friend.

Rosencrantz is the kind of friend I used to have – the kid who is constantly trying to manipulate people into telling him more than they want to. He’s like the friend who says, “If you were really my friend, you would. . .” and it’s either something he wants you to do or something he wants you to say and somehow you end up doing it or saying it. But then you feel bad – and you think, “Yeah, you know – maybe I’m actually NOT your friend, so no thanks. I won’t tell you that intimate secret, actually. No, no, you don’t get to hear about my innermost feelings.”
This is a particularly sneaky manipulation – the idea that by somehow NOT telling Rosencrantz he is presenting his own freedom, he own cure.
When, of course, there’s nothing wrong with Hamlet that a little revenge won’t cure.