How can current flow through a capacitor(open circuit)? Why is there a need of displacement current and why does it exist for a short period of time? Does the displacement current exist in medium between the plates? Why is there a need for charge to flow from battery's terminal to capacitor's plate?
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?
When a capacitor is under charging one of the plates (= the minus side) gets more electrons and the opposite plate (+) loses exactly as many. The force which changes the balance comes from the external source and metal wires bring it to the plates.
No current is needed across the gap between the plates. The external source both pushes from its one side and and sinks into its other side the needed amount of electrons.
If there happens to be some insulation material between the plates instead of vacuum there can be current inside that material during the capacitor gets charged. That's because the electron configuration in the material molecules changes due the increasing electric field between the plates. But those electrons do not move to nor from the plates, they stay in the insulator material, they only averagely move towards the plus plate. That effect isn't the displacement current, it's called polarization. It effectively reduces the distance between the plates so easily polarizable (=high permittivity) material between the plates greatly helps to make high capacitance capacitors.
The displacement current is an unfortunate name for a trick that mathematician J.C Maxwell made. He was writing a systematic compilation of all known basic electricity and magnetism theory. His goal was to present known facts in a general form which do not depend on any practical applications. That means vector differential equations in 3D space.
One thing was harmful: As a capable mathematician Maxwell knew that his equations had a hidden contradiction which someone sooner or later would bring up to make Maxwell and his work laughable. One day he got an idea: The contradiction could be fixed by inserting a new reason for magnetic field: a change in electric field.
Before that day theoretical writings stated "there's magnetic field around current and there's no other reasons for magnetic fields except fixed magnets". They could also be seen caused by current in the structure of the magnetic material, but that was only guessing at that time. Maxwell inserted a term which made magnetic field also existing around a growing or decreasing electric field. So, there should be a magnetic field between the plates of a capacitor during charging or discharging just like there were a current in the gap between the plates. The calculated equivalent current between the plates is equal with the charging or discharging current.
Maxwell called that equivalent current "displacement current". A part of magnetic field between the plates can be caused by the polarization current in the insulator material and that's real current, but if there's no polarizing material between the plates, the magnetic field between the plates is generated by the change in the electric field and the equivalent current for the same field is purely imaginary.
Not asked, but maybe interesting: Maxwell himself didn't make any practical experiments to test how true his new law was. He knew that it was impossible to measure it with current equipment. There was no big enough capacitor nor measuring instruments compact and accurate enough for smaller ones.
Maxwell published his work. The readers took it seriously because Maxwell already had earned his spurs in the world of science. It took years before one physicist was able to show that Maxwell's equations really predicted some observable phenomenons better than Ampere's and Faraday's theories did before Maxwell's correction. That phenomena was radiowaves and the physicist was H.Hertz. Later others found that also the basement of the theory of relativity was included to the same equations.