Could the Third Brother be the East Wind brother? Sherlock never said "A story Mycroft used to tell me" which to me would make more sense; John knows Mycroft. And then Sherlock adds "He wasn't a very nice older brother" a weird detail to add considering that John knows how the two siblings are to eachother, and odd to me, considering that Mycroft is rarely ever outright mean to Sherlock. They have a rival certainly, and children do often grow up different than adults, but the detail just doesn't fit to me.

Then add on the fact that Mycroft is a guide of sorts in Sherlock's mind palace. That doesn't seem like the type of role you would give to a brother that tormented you as a child. Rather it seems like the role that you assign the older brother that always told you what to do and was usually right, having the advantage of the wisdom of age. Hence, the rivalry, Sherlock likes to be right and Mycroft is his biggest challenge.

Maybe this other brother (Middle or Eldest, since we know that this brother is older than Sherlock, but not whether he's older than Mycroft) is a brother that they grew up with. (Middle seems more likely to me, old enough to torment Sherlock, but not old enough to be able to touch Mycroft. It would also explain the age difference between the two brothers. We know that their mother left her profession to have children, as they weren't likely accidental it seems unusual to have children that are that many years apart. Stick another child in the middle and it seems much more likely) Maybe he's a really Psychopath, unlike Sherlock, perhaps that's why Sherlock is always so touchy about people calling him a psychopath, even though usually he brushes off what people say about him (though I admit that is reaching, and he probably just wants to be right). 

So Mycroft does something to protect Sherlock. Who knows what he does. Maybe none of their family knew that Sherlock was being tormented and Mycroft finally discovers it. It'd be a great explanatio