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.
Ultimately, one cannot be held responsible for how people react. However, there are good, indifferent, and bad ways to part company that do influence how people might react.
First and foremost criticise behaviour not the person. I really cannot stress this enough. It is there player's actions by lying about their age and not heeding to the warning that brought you here. Not that they are an immature jerk. They key here is to tell the truth, as it is. Avoid strong antagonist words, instead use more neutral one, for example do not use "kick out" but use "part ways". Do emphasize those behaviours that are not inherently bad but just did not mesh with the current group. This is a nice way to lead into my second point.
As david-k pointed out in a comment:
The conversation could say "You like to play this way - which is not at all a bad thing - but it doesn't fit well with the style we've chosen for our group. You might find that Group X will fit your play style better."
Secondly, offer advice about the type of games and groups this person should be looking info. Even offer a reference letter -- if such things exists in Roll20. This is your opportunity (while remaining truthful) to make said player feel better. Rejection is hard to take, this makes the pill easier to swallow.
While they are early teen, do treat them like you would an adult. No matter what, do not patronise them. Speak to them like you would like your significant other to break up with you: honestly, compassionately, and empathically. Rejection is best handled face to face, in person or using video. Any other ways is sub optimal: voice is meh, email is bad, text is worst. Since OP commented that video is a no go since the player wisely does not want to share personal data, then voice is the best alternative. However, be aware that you will be missing the vast majority of social cues thus you need to be even more careful of how you say what you need to say.
Whatever medium, do prepare what you are going to say (the method of loci is great) and rehearse it, maybe even run it past someone your trust.
Finally, say that it is okay to part. Wish them luck in their next game and leave the door opened for future interaction -- only if you wish some.
As a side note, early teens are hard years for anyone. Do check with them that they are okay in a few days.
You could follow the advise of ignoring the problem and finishing the game. I find it despicable advise because it is manipulative, harmful in that it stops someone from learning from their mistakes, and will almost certainly feel like a betrayal were it ever found out.
Best Answer
You could try using the "Same Page" tool. Go through the questions with your players as a whole and come to a consensus on what type of game you want to play. Keep in mind the tool is not a survey, but is meant to be gone through as a group and finding answers to each question in the tool is important to find the sweet spot for your particular group.
Ultimately, you need to have a candid discussion with your group about what they are looking to get out of the game. Sometimes people just want to come and hang out with their friends for a few hours and don't otherwise care. Some people are really into their characters and want to role-play everything, while others just like to watch their numbers get bigger as their characters increase in power. In my experience, disparate groups with wildly varying expectations (or very few or no expectations) tend to veer towards the dynamic that occurs between the GM and the most "powerful" player. In other words, people with big personality who tend to take control (even when they don't need to) tend to steer the group dynamic.
It can be difficult to deal with this, especially when you as GM may want a more RPG-heavy story while your "leader" player just wants to power game his way through every encounter (of course the same thing happens no matter what the difference is -- this just happens to be an example from my own experience). Other players will tend to follow the player leader in those cases, and you might find yourself running a different game than the one you thought you were running.
Don't forget that you, as GM, are also playing the game, and your enjoyment also matters. If you aren't having fun because your players don't want to play the same way you do, then you may need to mix it up. How you resolve this problem isn't always easy -- it might even involve dropping your group or players from your group. It could also mean simply adjusting your own expectations so that you aren't let down when the gameplay doesn't go the way you want it to. This has been the best way to deal with these types of situations in my own experience.