Now let's assumme it has Double Strike...when it deals damage, do i take 2 extra turns? or does the clause "affter this one" resolves in taking twice the same extra turn.?
Yes you will get two extra turns since it hits twice it will trigger twice and add a turn into the cycle after the current one twice. after this one only makes it so the last turn added is the 1st one to be taken, the other one is still there just goes 2nd.