A force opposing Gravity Every Action Has An Equal and Opposite Reaction (Newton's Third Law.)
If this is the case, does gravity have an equal-opposing force? 
From asking around I still haven't got a very clear answer; those who I've talked to seem to believe there isn't one - that gravity is actually a singularity [a one way force] which somehow "just works", others think it differently - believing there is an opposing force of which prevents gravity from compressing masses more than it already does.
So which one is the right answer? (if either!)
 A: Yes, every gravitational force in Newtonian mechanics has an equal and opposing force, and it usually acts on other mass. 
More specifically, every two pairs of masses feel a gravitational force that's proportional to the product of their masses and inversely proportional to the square of their relative distance, but more important is the fact that both masses feel the attraction to each other.
Thus, when you throw a ball of ~100g in the air, it experiences a gravitational force of 1N downwards, and in doing so it exerts a force of 1N upwards on the Earth. The reason you don't observe the Earth moving is that its acceleration is so small (on the order of 10-25 m s-2) that it gets swamped in everything else, but it does happen.

Now, it's important to note that gravity is not usually the only force acting on any object at a given time. If it is, then the total force will be nonzero and the object will accelerate (as per Newton's Second Law). Conversely, if an object is not accelerating, then the net force on it is zero, and there must be additional forces that cancel out the gravitational one.
For a book lying on a table, for example, the weight is cancelled by the upwards reaction force from the table. (And, of course, this gives an added reaction force downwards from the book on the table, which gets cancelled by a correspondingly larger reaction force from the floor on the table.)
Similarly, the reason that masses (like, say, the interior of the Earth) don't get compressed any further is that any given volume of rock will be acted on by the downwards gravitational force and by the upwards pressure from the rocks below it.
A: My knowledge is limited on the subject but matter is typically prevented from collapsing under the weight of extreme gravity by particle degeneracy. This is what keeps neutron stars from collapsing into black holes and is the result of particles resisting occupying the same quantum states. 
There are also some recent observations that indicate that there is a repulsive force that acts only over large distances (almost the inverse of gravity) causing the universe to expand at an accelerating rate. Why this occurs is still a matter of debate and there are various theories from the topography of space itself causing gravity to act repulsively to an as yet undiscovered force causing this.
A: Suppose you're standing on a box as shown in (a) below:

There are four forces acting. You apply a downward force $mg$ on the top of the box, and by Newton's third law the box applies an upwards force $-mg$ on you. The box transmits your force to the ground, so the box applies a downwards force $mg$ on the ground and the ground applies an upwards force $-mg$ on the base of the box.
So far so good. But how suppose we suddenly pull the box away as in (b). There is still a downwards force $mg$ on you, and indeed that force is going to make you fall downwards. The question is whether there is an equal and opposite force upwards.
The answer is that yes, there is indeed an equal and opposite force of $-mg$ on the Earth, so Newton's third law still applies. The confusion arises because we normally think of the action and reaction force as operating at the same point. So in (a) there is an equal and opposite pair of forces at the top of the box and another equal and opposite pair of forces at the base of the box. But now we seem to have the action and the reaction separated in space.
With Newtonian gravity you just have to accept that there is a gravitational field in between you and the Earth, and this field transmits the force on you to the ground and the force on the ground to you. To really understand what's going on you need to understand general relativity. This tells us that the Earth curves spacetime and this creates the downwards force on you, however your mass also curves spacetime and this creates an upwards force on the Earth.
A: 
For a book lying on a table, for example, the weight is cancelled by the upwards reaction force from the table.

That's not quite true -- unless the table in a vacuum chamber at the south pole. The upward normal force exerted by the table and the downward gravitational force exerted by the Earth don't quite cancel. The book rotates with the Earth, and except at the north or south pole, this rotation means the book is accelerating. It's a rather small acceleration, about 0.35% g at the equator, but it's not zero. This means the net force on the book is not zero, and that in turn means the weight of the book is not quite cancelled by the table. What about the vacuum chamber? The buoyant force of the air on the book is about 0.15% g. This means the upward normal force on the book by the table is about 0.5% less than the book's weight (at the equator).
Newton's third law says the forces in an action-reaction pair are equal but opposite. Forces that are only approximately equal and approximately opposite do not qualify as a third law action-reaction pair.

... others think it differently - believing there is an opposing force of which prevents gravity from compressing masses more than it already does.

Gravity can compress masses quite a bit. The density of the stuff (mostly hydrogen) at the center of our Sun is about 150 times the density of water. Closer to home, the density of the stuff (mostly iron) at the center of the Earth is about 13 times the density of water. That's about 2/3 greater than the density of iron at the surface of the Earth. That increased density is because the iron in the Earth's inner core has been compressed by quite a bit. The rock at the core/mantle boundary similarly is considerably denser than is the same rock at the Earth's surface.
The pressure from below is not what counteracts gravity inside the Earth. What counteracts gravity is buoyancy. Imagine a chunk of rock deep inside the Earth. The pressure at the top of the rock is slightly less than is the pressure at the bottom of the rock because of hydrostatic equilibrium. This pressure gradient results in a buoyant force that keeps the chunk of rock where it is.
What stops pressure from compressing a chunk of iron or rock inside the Earth into nothingness is a quantum mechanical effect. The electrons get pushed ever closer together with increased pressure. Those electrons can't share space thanks to the Pauli exclusion principle. The material does compress with increased pressure, but this compression stops at the point where the quantum mechanical pressure balances  the external pressure. 
A: Assume a person is falling towards the earth. We know that there is a force and thus an acceleration acting on the person. The opposing force is the gravitational force exerted by the person onto the earth equal in magnitude (Newton's Law of Gravity). This force produces an acceleration (Newton's Second Law) but because the mass of the earth is massive as compared to the human body, it is relatively small and can be taken as zero . Adding these two forces gives a net resultant of zero, (Newton's Third Law).
