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.
The consequences of the wording of a wish are, as you note, entirely at the whim of a DM. No one can answer how any particular DM will rule, so keep in mind that any answer is on shaky ground, subjectively.
A DM could rule that you are granted this ability by virtue of being unable to cast Wish for anything but spell duplication, for example.
However, the wish you specify will have to be cast before it takes effect, and you could almost certainly undergo stress from it, with the potential of it being permanent.
That would be very powerful.
Let's compare the effect with the other things Wish can do:
These are all temporary or one-time effects, excluding the resistance. Permanently being able to cast all of his spells at will would be way too powerful; refreshing all of the Wizard's expended spell slots, on the other hand, would be more balanced.
If you want to deny his Wish, you could twist his wording and make it so that he can cast all his spells at-will - but only at some poor bloke named Will.
Alternatively, you can simply interpret it so that the Wish has (mostly) no effect at all, if you say that "he can cast all his spells at will" means that he's able to cast all his spells at Will. Since he presumably could already do that before, it changes nothing. This interpretation would also not include any at-will casting.
Another interpretation is that he can target Will with all of his spells, even those that normally only have a range of self or require a willing target (for example). I would, however, be very careful with this interpretation, especially concerning the willing target - many spells become significantly more powerful if their target doesn't have to be willing (although I can't think of an example off the top of my head).