Let me tell some stories:
1. The negotiations with Hector Valois
In this mission, my PCs were tasked with bringing peace to a group of nations that were beseiged by what I described as a Dragonlord. Hector Valois was sending his minions out on raids, claiming territory and assaulting villages with impunity. Had the PCs fought him on day 1, in his own domain, he would've slaughtered them. Instead, the PCs decided to negotiate a peace summit between the 4 nations and Hector. Along the way, they decided that Hector was both likely to negotiate dominion over the combined territories, but also a tyrant. Determined not allow this outcome, they searched far and wide, finding the oldest Dragon (a Gold) alive to supplant Hector. With this Gold Dragon in tow, picked a fight with Hector within the summit's main event.
The PC's "team" consisted of a few volley-archers off-screen, a CR 16 Gold Dragon, and 3 ECL ~8 PCs (my ruleset is E6+Gestalt in 3.5) against Hector (CR 10+), and 3 young adult dragons (CR 9/11/13 respectively). The PCs were HOPELESSLY outmatched for the fight they took, but then again, that's the fight they CHOSE to take.
They could've lured Hector into an ambush. They could've picked a fight when he didn't have his dragons in support. They could've organized a larger response to him. etc, etc, etc.
2. The unfair trade with the Wizard Dominic
Another group of PCs I was gaming with have an interesting story. One day, a PC was cursed/wounded/whatever, and required a real cleric's aid. The PCs decided that their best course of action was to find a cleric who was wanted on the bounty boards and get that cleric's help. They found and made a deal with this necromancer, and then instead of paying him, attacked him. The necromancer escaped. Later, the PCs manage to hunt him down, but in the process of getting close to him, the Necromancer unleashes a terrible retribution: He sends his minions (Wights) to wipe out their favorite town.
The PCs learning this, then decide to try something clever. They contact Dominic, a wizard of GREAT resources and power, and offer him a deal, they'd surrender to his demands of them, if only he'd save this poor town. Dominic agrees, and the town is largely restored, but price proves to high for our plucky PCs.
Reading your question, its my assessment that you would feel attacked by me if I was your DM. Partially, this is because even in your own retelling, none of the other PCs are objecting to the DM's behavior. Partially its because these actions seem mostly reasonable, except for your value-judgements of them. I believe that the correct course of action for you is to start assuming that your DM is both A) acting in good faith and B) acting with a degree of skill such that you might learn something from him if you ask WHY he's running the game the way he is.
Essentially, my advice to you is simple: Talk to your DM. Make it clear that you are not having fun, but DON'T characterize your lack of fun as his fault. Let your DM discuss the whys with you. And if he gives you actionable feedback, act on it.
Not your fault. It sounds like the rogue's player doesn't really get that it is a game and/or that he has not really grasped that he shouldn't act like a child anymore. He clearly wants to control, to pause the narrative to argue and complain to his own character's advantage. The word that springs to mind is "petulant".
That doesn't really help your group except that you personally should not feel bad, it is everybody's responsibility to behave in a way that supports a fun game and narrative and it doesn't look like that player is keeping up their end of the bargain.
He is not understanding the difference between the players and their characters, the difference between players talking together (telling the story) and characters talking (part of the story).
I would suggest that the point at which you lost a lot of your, errr, authority over the situation was the point at which you backtracked and allowed the rogue's actions to effect the past:
And then it all broke down. The player who plays the rogue wants to stop the monk from touching the rings. I make them do a dex throw and the rogue won. But I had already said the rings disappeared, so in order to keep the narrative going and not allow them to meta-game with the knowledge, I ruled that the rogue was able to stop the monk from taking the ring, but he touched it with a finger and the other two disappeared.
You have already identified this as the point at which "it all broke down". The monk had already succeeded and "The other players did not react", so here you succumbed and let him in.
The one thing I (all of us by agreement actually) am quite strict about is once something has happened it has happened. If it is unfair, sometimes that's life, sometimes it is "repaired" by other things happening... but we never go back once is has happened and we have moved on, even when it is a clear mistake. We write it into the story and move on, and we all have an equal responsibility to accept it and gloss over/ignore any issues and not worry too much about it. If we haven't moved on yet then there is some flexibility, but in your case you waited for a response and then the other rings disappeared, you had moved on and he wanted to change the past because he was to scared to make the first move and then lost out.
They trust me, and each other, enough to not do anything deliberately unfair that is not an expected part of their characters or the set up of the game, and they respond respond to in-character things in-character, and don't take things too personally. They also trust me to be very fair over the whole game in terms of opportunity and to listen at an appropriate time if someone is not enjoying things and try to make it enjoyable again, which is, after all, the point. Even the rules aren't that important, apart from to allow players to have realistic expectations regarding their character's actions.
Setting this situation up takes work by all, not just you. Looks like you have already spoken with him about it and have come to an impasse. The bottom line is this:
"He starts making threats, saying he cant play with us if we can't play like he wants."
If the rest of you don't want to play like he wants, which sounds like a un-attractive way to me, take him up on the offer...
By the way this is a very typical response (seen it before, done it myself) for someone who is taking it all too seriously and can't deal with the feelings and consequences of their actions when it goes wrong. Again it is not your fault. When my children do it, it's called a tantrum and is all very excusable and understandable (if inconvenient and sometimes upsetting) as they aren't mature enough to deal with what they are feeling. It's made better not by giving into their out of control emotions (that just leads to more of the same, through positive reinforcement, which is not good for their emotional development), but instead by making them feel safe and allowed to be upset and have the feelings, but standing firm in the face of them. it means they develop some control and are better able to make decisions about their behaviour. It's a different story and expectation when a twenty-something adult behaves this way and they are often not open to an emotional intelligence based discussion, nor is your relationship with them often one that makes it appropriate to counsel them.
Sorry about the long winded answer. As you may read into what I have written I've encountered this and thought about this quite a lot and have a precarious balance between compassion and lack of patience for adults who don't have the self-awareness to do anything about behaving this way. And I'm not innocent of it myself, though hopefully I left it behind in my early twenties.
Best Answer
"I warned everyone beforehand that death IS a possibility in my games"
Then you lied. Sorry, that might sound a bit harsh. What I mean is that you warned them that death is a realistic possibility, but you have planned for scenario where death is guaranteed, or at the very least, is the cost of doing business with you - one of you must die to keep the game moving forward. "You might die" and "I am going to kill at least one of your characters" set up entirely different game expectations. The former tells me to expect challenging, high stakes encounters that I can overcome with some ingenuity and luck, but "I'm going to kill one of your characters" tells me to expect you to present an insurmountable challenge. You said "you might die", but what you meant was "you are going to die".
You write:
This is just not the way to do it. For two reasons - the first outlined above. The players are expecting challenges, and you intend to present them with an impossible dilemma, which, in the moment, is likely going to be poorly received.
Your plan will probably send the opposite message.
The second reason: you say you want to do something to remind them that death can happen, but artificially forcing death upon them only to reverse it by saying "Surprise!" takes away from the stakes you are trying to present. Your plan does not say "the stakes are high, the challenge is real", it says "the stakes are low, the challenge is fake".
Keep the stakes high, and deliver on your promise.
Instead of killing a character just to Deus Ex Machina them back to life, make the puzzle solution costly. I've done something similar in a dungeon crawl, but I didn't kill them, I just increased the challenge. The party was met with a door and a riddle, with a similar solution - blood. Once they figured out the narrative solution to the puzzle, blood, I explained to them the mechanics. They had to sacrifice hit dice to unlock the door. The party of four 8th level characters had 32 hit dice between them, and the door required 16 hit dice sacrificed. The important part of this solution is that the party could strategically decide who would give more and less hit dice. There was a price to pay, but they had control over how that price was split between the party members. They opted to let the barbarian with the most valuable (d12) hit dice keep all of them, with the expectation that the barbarian would be taking the most damage as the frontline tank moving forward. The party felt like they still had control over their destiny, despite having to pay a price to advance, and the price increased the risk of death moving forward, but it did not guarantee death. In your situation, they are going to go into it feeling like they have little control: "one of you has to quit playing or we all quit playing".
I'll leave you with this guidance from the intro to the Dungeon Master's Guide:
This encounter as you describe it does not "revolve around their actions and decisions", it revolves entirely around you forcing your preplanned outcome, an outcome that at first looks exactly like slaughtering adventurers.