Would the ‘net’ flux through the area bound by the coil be zero?( because of opposite directions of magnetic field at its two faces?)
The magnetic field actually is in the same direction on each side of the plane. You can see this in the image below and as explained here. As you can see, there is definitely a magnetic flux through the area bound by the coil itself.
If so, Gauss’ Law states that net flux through a closed surface is zero, but the area bound by the coil is not a closed surface.
Yes, you are right. Gauss's law for magnetic fields tells us
$$\oint\mathbf B\cdot\text d\mathbf A=0$$
but this is a surface integral over a closed surface. The area bound by the coil is not a closed surface, so we don't need to worry about this applying here.
So, even though the flux through thiis area is in fact not $0$, I will address a concern you seem to have in linking these two ideas together. You seem to be thinking that a $0$ flux means that the surface integral must have been done over a closed surface. This is not the case. The statement "If the integral is over a closed surface, then the magnetic flux is $0$" is not a biconditional statement. In other words, the statement "If the magnetic flux is $0$ then the integral was done over a closed surface" is a false statement.