Is it possible for an object to have mass but zero volume? Can there exist a particle/object in the universe having mass but no volume? Is it possible that mass can exist without volume and density? We think we know that matter is anything having mass and that it occupies space, but is it possible that this statement is wrong?
 A: Your question states that

We think we know that matter is anything having mass and that it
  occupies space

but in fact, we know better than that. We have good reason to believe that fundamental particles are point-like. In other words, they have no internal structure, size, or volume. And they indeed have mass. We have a theoretical understanding (in local Quantum Field Theory) and experimental evidence (from collider experiments) for objects with mass but no volume.
This isn't the final word, though, because it's quite possible (and some might say very likely) that particles that appear point-like in our experiments have substructure, including a characteristic size, that would be revealed in very high-energy experiments. I believe that to be the case in string theory.
In summary then, we have no evidence that particles are not point-like and a solid theoretical understanding of point-like particles, but there is motivation for considering internal structure, which could be revealed in very high-energy experiments.
A: In the standard model  of particle physics which fits the data up to now elementary particles entering the lagrangian are point particles with mass.
The electron, for example is one of the elementary particles, and it does have a mass and the fit gives it 0 volume.
There are experiments which try to set limits to how small the volume of the electron is. The fact that the standard model fits a large number of measurements in elementary particles with zero point dimensions for the elementary particles can be considered as a measurement.
One should keep in mind though that the electron is a quantum mechanical entity , and follows quantum mechanical equations. If one tries to determine the size the probe that checks the size follows the Heisenberg uncertainty principle, which means to get very small limits one needs very high momentum probes.
String theories posit that elementary particles are vibrations on one dimensional strings, with a length smaller than the Planck length. Still, a line has no volume, even if this is so.
A: I'll address your question a little different, because talking about volumn and particles is problematic in many ways.
Let's phrase your question "can there be two particles with mass be at the same place".
The answer is yes. There are two types of particles:fermions and bosons. While fermions (electrons, protons) repel each other (not only because of the charge!) Bosons can be packed as dense as you want.
For example photons are bosons, but even fermions can be coupled to form bosons, like the helium atom.
You see, this is a little different from your question since helium as clearly a volumn but photons don't.
A: Based on the latest breakthroughs in particle physics, the answer is a plain NO - it's not possible for a massive particle to have no volume. In fact, it is NOT possible for any particle, whether massive or massless, to have zero volume.  ALL particles have a certain volume, no matter how small beyond observation.
On the contrary, mass is an intrinsic property of a particle that arise from the Higgs mechanism.  Particles that do not interact are massless, those that do exhibits mass.
However, as research continue to grow and our knowledge base is updated, the answer can change, or be uncertain, much like the uncertainty principle. =)
A: It is not possible for any object to have mass but zero volume, because in order for this to occur there would have to be infinite gravitational field around the object, and this does not happen in the natural world (I will come to the notion of "point particle" in quantum physics in a moment). When mass is concentrated in very small volumes what happens is that a black hole forms. The heart of a black hole is a region where our understanding is limited, but in any case a black hole has a non-zero surface area, the area of its horizon.
Quantum physics is built out of a physical and mathematical set of ideas called quantum field theory. Quantum field theory may itself arise from a yet more basic set of ideas called M theory or string theory. In both quantum field theory and in string theory the stuff which makes up the natural world is quite subtle and there is no simple straightforward sense which one could say the universe has point-like particles in it. Rather the term "particle" is being used in a rather technical sense, as a way of referring to a part of the complete description. In the complete description there never arises a concentration of a finite amount of mass with an infinite density. This is partly because a black hole would form, but more commonly it is because as soon as one sets up a situation where the mass is concentrated in a small volume, one also has large amounts of momentum (because small volumes require small wavelengths) and hence large amounts of kinetic energy. This kinetic energy will in turn make it possible to generate particle-anti-particle pairs. So this is what happens. A classic example of this is seen in all high-energy physics experiments involving particle colliders. There you have an attempt to focus energy (and therefore also mass) into a small volume at a collision vertex, and what happens is not huge density but huge amounts of pair creation.
When a basic particle such as an electron moves along in otherwise empty space, there is no concentration of mass in an infinitely small volume because the wavefunction of the electron is spread out. The very term "electron" or "particle", in quantum physics, refers to this thing with a spread-out wavefunction.
A: 
Is it possible for an object to have mass but zero volume?

No. 

Can there exist a particle/object in the universe having mass but no volume?

No. 

Is it possible that mass can exist without volume and density? 

No. 

We think we know that matter is anything having mass and that it occupies space, but is it possible that this statement is wrong?

Yes. 
Let's start with an E=hc/λ photon. It has a non-zero energy, and a non-zero wavelength. So it's a wave, not a point-particle. Now have a look at wind waves on Wikipedia. See the gif with the red dots, and crop it to remove the surface and emulate a wave in space:
 GNUFDL image by Kraaiennest, see Wikipedia Commons 
Now think about pair production where we create matter from light. Then remember electron diffraction and the wave nature of matter. And most important of all, remember that it's quantum field theory, not quantum point-particle theory. The electron is not some speck that has a field. Instead field is what it is. And this field doesn't have an edge or a surface. It gets weaker away from the centre, but it doesn't stop, ever. The electron just isn't a point particle, despite what some people say. Nor is a proton, and nor is a hydrogen atom. But a hydrogen atom is just an electron and a proton, and they are "just their fields". So the hydrogen atoms's gravitational field isn't something distinct from the hydrogen atom. It's part of what it is. The same applies to a star, and to a black hole. 
So there are no zero-volume masses.   
