Current through capacitor, displacement current
How can current flow through a capacitor(open circuit)? 


Because displacement current "flows" between the plates.  Or, more simply, because electrons pile up on one plate and are depleted on the other.


  Why is there a need of displacement current


There is no "need" for displacement current, it just is.  Or, more properly, in order to make his equations consistent, James Clerk Maxwell invented it so that Ampere's Circuital Law always works.

You don't need the concept of displacement current to describe how a capacitor works at a simple level -- you can just use electrostatics and the notion of charges piling up on the plates.  But to keep Maxwell's Laws consistent, you need it.


  and why does it exist for a short period of time?


Define "short".  If it takes a day for a capacitor to charge up, then the displacement current will exist for a day.


  Does the displacement current exist in medium between the plates?


Yes.  That's the point of displacement current - it makes Ampere's Circuital Law work (and by extension, lets us calculate the propagation of electromagnetic waves).


  Why is there a need for charge to flow from battery's terminal to capacitor's plate?


How could there possibly be a way for charge to not flow from the battery's terminal to the capacitor's plate?  If there's a force pushing electrons onto one plate, and an equal force pulling electrons off of the other, wouldn't you expect them to move?