How much power a child or teenager ruler had of his/her own (in rl and in ASOIAF). I know that in ASOIAF rulers had councils and I suppose that the same extends to rl, but in the end, who had the last word when deciding about rulership, if the young ruler made a public proclamation without consulting the council, do people was supposed to take it as a fact, or there was some kind of unspoken understanding that because h/s was underage didn’t have the “authorization” to make desicions of h/h own

Well, this is the fly in the ointment: no, the underage ruler’s proclamations aren’t supposed to be taken as law. The whole idea of a “regency” is that the underaged ruler is not, in fact, the ruler until they attain their majority. 

So historically speaking: no the young ruler would not be taken seriously if they tried to go ahead with say, the execution of Ned Stark, without the regent/s’ consent. The only reason it worked with Joffrey is A. collusion ahead of time by Littlefinger and Janos Slynt, and B. Cersei is really, really bad at improv.