Well, to start with my own analyse on this which does not sound odd,
The sentence can be broken into pieces to make it more understandable, we use accusative case to indicate the direct object. Меня зовут Çağatay (They call me Çağatay) who is called here? Çağatay, for sure. So the "who calls" part is in Nominative.
Твои братья зовут тебя, who is called here? "Ты" which is тебя in accusative case. Who is calling? Твои братья.. nominative.
Let's reverse the example as you are calling your brothers.. ты зову своих братьев.. as you see it in accusative plural..
Well to be honest it sounds more easy to understand when I refer it to Turkish language but for English learners it may be tougher to figure out (Russian cases)
Does братья translate to both brothers and sisters (for example a group consisting of a mixed bunch of men and women), or it's only used when referring to a group of males? I wonder if it's like spanish, where you usually call a group in its masculine form, no matter if it's conformed by a mixed gender.
to ring - звонить. to be called - называться. Here are the meanings: http://translate.academic.ru/%D0%B7%D0%B2%D0%B0%D1%82%D1%8C/ru/