Gravity at the centre of the Earth At the centre of the Earth, if there is zero gravitational effect due to the Earth's mass, would the Sun's and the Moon's gravitational effect still be felt?  So does the liquid centre get pulled towards the the surface facing the Sun or Moon as the Earth rotates?
 A: As the gravity due to Earth is zero at the center, the mass at the center will continue moving along with the Earth's center around the sun, i.e., it will be in orbit. That is the effect of Sun's gravity.
As to the moon's effect, just like tides, the mass will suffer small fluctuations in position, if left totally free. In which case it will oscillate, and as soon as it deviates from the mean position, SHM motion due to Earth's effect will be added to it's motion.
Taking into account all the forces in the question, this is what I think of the final motion:
the object will revolve around the sun in an orbit (alongwith the Earth) and execute SHM with mean position varying due to the effect of the moon.
A: The earth is in orbit around the sun, so the sun's gravitational pull isn't felt anywhere on the earth, because in effect we're continuously falling towards the sun. 
A: If Earth is really composed of concentric homogeneous spheroidal or ellipsoidal shells, then, the total Earth's gravitational field at its center is null. It is know in stellar dynamics as 1st and 2nd Newton's theorems. It is also valid for stars, spherical galaxies, or any other object with this type of mass distribution. 1st and 2nd Newton's theorem can be seen in Binney & Tremaine "Galactic Dynamics".
Sun gravitation affects Earth movement, of course. Earth acceleration is a = GM/r², where G= Gravitational constant, M= Sun Mass and r = distance Sun-Earth. all the objects on Earth feel the Sun gravitational field as well. Nevertheless, Earth different parts are pulled differently due to the tiny difference in distances between each one of these chunks and the Sun. These differences produce the Sun tidal force on Earth. Similar effect produce the Moon. The tidal acceleration is proportional to the Earth diameter D, an inverselly proportional to the distance Earth-Sun to the 3rd potence, D/R³.  
A: INTRODUCTION
Gravity at the centre of Earth is zero because the Earth is symmetric to its axis.
SPECIFICATION
OK here is the important thing - you need to specify reference frames. That is, if you view it in the centre of Earth, then of course the gravitational force exerted by other celestial bodies cannot be felt. This is due to the Earth's centripetal motion around the Sun (and orbit fluctuation with the Moon) which balances the gravitational force between them.
EXPLANATION
If you view it from space, for example, all three celestial bodies mentioned above are attracted by each other due to Newton's Theory of Universal Gravitation. That is to say, if there exists an absolute reference frame, which allows the whole universe to be observed, any matter with positive mass will experience mutual gravitational pull, regardless of whether it is at the centre of mass or not.
CONCLUSION
Simply put, anything we know of already (positive mass) will certainly experience mutual gravitational force. However, it does not mean the iron-cobalt core of Earth will get pulled towards the celestial bodies because their convection motion and self-spin offset the forces of attraction.
