No. Your behaviour wasn't right and yes, you should have apologised for what you said.
TBH I can't even believe you're asking.
He was constantly sniping at me, having a go at me about a word I had said by mistake, constantly going on about it, making me out to be stupid and thick, so in the end I said "you're lucky I don't sock you straight in the mouth for saying that". He didn't respond at first but then started staring at me in a sarcastic way so I said "Carry on looking at me like that and i'll smash your f*ing teeth right down your throat".
This paragraph illustrates just how wrong - and unnecessarily dramatic - you were. You said something wrong. You were laughed at for it. And this is the bit that's key. You don't like being laughed at or made out to be wrong IN ANY WAY. We see this in all the threads and posts from you.
Rather than letting it lie, which an adult would, you responded in a totally unnecessary way. In your own words, too, you point out that he didn't actually say anything in response to your first verbal attack - yet you responded with the threat of violence to a look from him.
Can you not see that this is a pattern of behaviour from you? You take offence very easily, perhaps when none has been intended, and then blow up in this kind of dramatic fashion.
Then you complain bitterly about being called out for it. Worse, you go on to make it all about you! When told that your boyfriend's brother was upset about your threat (and who wouldn't be?!) you say:
I don't actually believe this, I feel like it's a ploy to drum up sympathy for himself and to make me look worse
It's all about you, yet again.
So yes, you should definitely apologise, for a reaction and a comment that was totally out of order and proportion.
And have a good hard look at yourself and just THINK about all of the things that you post here, and the responses you get from other posters.