If a constant current can create constant magnetic field , why do we need changing magnetic field to induce current? 
The electrons only going from negative to positive are able to create a magnetic field, why does the magnetic field have to keep changing to induce a current? If I pass the same magnetic field through the coil, why doesn't current keep flowing? 
 A: One answer to your why, is, because that is what Maxwell's equations say,  and they model perfectly the data we have of classical electromagnetic interactions. 
Particularly the form :
$$ \nabla\times E = -\frac{\partial B}{\partial t}, \;\;\text{Faraday's induction law} $$
The change in the electric field in space is affected by the change in the magnetic field in time. As current happens only when there is transfer of electric charge, if the time derivative is zero then the electric field that could move electrons or ions to generate a current, would be zero.
Note that it is a law, i.e an axiomatic statement, derived from observing data.
A: A magnetic field produces current in a wire as it pushes electrons in a certain direction until the electrons compress and their electrostatic repulsion counters the magnetic driving force; equilibrium is reached, there is no current. A wire within a constant magnetic field will always reach an equilibrium state in which there is no current in the wire. Since equilibrium is reached so quickly, we see that a current can only be achieved when we change the magnetic field, if we keep changing the strength of the field, we change do not allow the system to reach equilibrium, thereby maintaining a current. 
This is entirely different from the reason a current creates a magnetic field which is due to relativity. While intuitively they may seem related, the facts of changing magnetic fields producing current and a current producing magnetic fields have nothing to do with each other.
A: 
If I pass the same magnetic field through the coil, why doesn't current keep flowing?

As you introduce the magnetic field to the coil the magnetic field will be changing. Thus an initial current will be started.
The current will stop flowing because of the non-zero electrical resistance of the coil/circuit.
I believe that superconducting coils will maintain the initial current indefinitely, but I have no personal experience with that.
A: It is a common misunderstanding that you think the magnetic field does not change. Even if the magnetic field does not change in space (its position), it changes in the time dimension.
This change of the magnetic field in the time dimension is what is needed to induce the current.
