Will a stone thrown follow an elliptic trajectory if it is allowed to pass through Earth? Normally, it would be a parabola. But I was thinking what it would be if were to assume that Earth (and its atmosphere) offer no resistance to the stone's fall.
 A: The parabolic path that we get for projectile motions  of rocks and other things is an approximation assuming that the gravitational field is of constant strength and direction.
If we truly considered the gravitational force as $\dfrac{-Gm_rM_E}{r^2}\hat{r}$, where $r$ is the distance from the rock mass, $m_r$, to the center of a spherically-distributed earth mass, $M_E$, ($\hat{r}$ points from the earth toward the rock at every instant), the path of the rock would depend on the total mechanical energy of the rock/gravitational system, $$E=\frac{1}{2} m_rv_r^2-\dfrac{Gm_rM_E}{r}.$$


*

*If E<0, the path will be an ellipse.

*If E=0, the path will be a true parabola.

*If E>0, the path will be a hyperbola.


The question takes on a large difficulty if you actually let the rock fall through the mass distribution of the earth. This makes the mass term, $M_E$ in the gravitational force a function of $r$ (because the mass "above" the rock no longer affects the rock in this spherically symmetric problem).  That additional $r$ behavior of mass causes the force to behave like linear restoring force rather than an inverse-square force.  That will change the orbit quit a bit and  I don't have time to work of the mathematics of the orbits.  Maybe someone else can provide the details in a follow-up answer.
A: Yes, its orbit would be elliptical, unless you project it in tangential direction(tangent to radius vector) with the orbit velocity at that point(its almost impossible to do so as there would be oscillations and it wont be exact circle) . In general it would be an elliptical orbit. 
However you can make it parabolic by projecting it with escape velocity(should not be directed vertically upwards).
Also it can be made hyperbolic if you throw it with speed greater than escape velocity.

I have ignored air resistance and gravitational force from other bodies. In basic kinematics we claim its path to be a parabola as normal projectiles rise to a very small height.
A: Yes.
The stone would move along a very-very long and narrow ellipse. Center of where-previously-was-Earth would be at the far end of this ellipse.
UPDATE: this is in case if all the mass is concentrated in the Earth center.
A: The stone will continue to move until the velocity of stone is opposed by air drag . If we assume air drag to be absent then stone will reach out of Earth atmosphere .And  swirl around in the universe
