Transistor activation / deactivation voltage in
You are getting wrapped up in matters that are irrelevant to the question.  The question states that when the voltage reaches 3.4V, the transistor is turned on -- it doesn't say that when the voltage falls below that the transistor is turned off.  So just assume that it happens by magic in another part of the IC, and answer the question at hand.


  If The transistor activates at 3.4 V, then when does it turn off (ed)?


At some time that is not addressed in the problem, through some mechanism not described in the problem.


  Does it deactivate at 0 V in - ~0 V in AFTER it has activated (opened/activated), and only activate at 3.4 V in?


That is both unknown, and immaterial to the problem at hand.