There's a hollow insulating sphere that is uniformly charged with charge $Q$ kept in front of a big insulating square plate uniformly charged with charge per unit area sigma kept in $y$-$z$ plane. The question is: is the electric field inside the sphere non zero uniform or zero, complying with the Gauss' law conclusion that electric field inside a conductor is zero. The solution which i have seen to this question says that electric field due the shell's own charge is zero but it does have field lines inside it due to the uniformly charged sheet. How is it possible? Doesn't it contradict Gauss Law?
There is no contradiction of Gauss's Law because Gauss's Law only constrains the net flux through any surface. In particular, it requires that a surface containing no charge have no net flux through it; this is a weaker condition than requiring that the electric field is zero everywhere on the surface.
In fact, any surface in a uniform electric field has no net flux through it. This is easiest to see in terms of field lines: since the field lines are straight lines and evenly spaced, any field line that enters a Gaussian surface goes straight through and exits the other side of the surface. Any negative flux from one part of the surface (where a field line enters) will therefore be cancelled out exactly by a positive flux on some other part of the surface (where the same field line exits), and the net flux is still zero.
Where your confusion may be arising is that if we have spherical symmetry, we can also argue that the electric field is purely radial, and has the same magnitude everywhere on the surface. Under this assumption, you can use Gauss's Law to argue that the electric field inside a spherical shell is zero, since if it wasn't, there would be inward or outward flux at all points on a spherical surface inside the shell, and we would have a net flux without enclosing any charge. But the symmetry argument is essential to this argument; and once we introduce the charged sheet into the problem, we no longer have spherical symmetry, and this line of argument no longer holds.
Yes electric line penetrate. It does not contradict gauss law because there will no electric field at centre only if charge is uniformly distributed over entire sphere. But in this case eventhough the sphere is placed near non conducting plates there will be coloumbic force between charge in shell and plate. Due to this reason the charges get pulled or pushed to a end according to whether charges in sphere and plate are equal or opposite. If they are equal then they repel and accumulated at one end . Hence uniform distribution is distorted and electric field is created..
The electric field inside the hollow, uniformly charged sphere in front of the uniformly charged insulator is, indeed, homogeneous and corresponds to the electric field of the charged plate. You can use Gauss's law separately to obtain the electric fields due to the charges of the sphere and due to the charges of the plate and then combine the electric fields.
(1) Inside the sphere the electric field due to the spherically symmetric charge is zero. This also corresponds to Newton's Shell Theorem.
(2) Inside the sphere the electric field due to the charged plate is constant and homogeneous.
(3) Thus the combination of both fields obtained from the separate applications of Gauss's law is the homogeneous field of the charged plate.