Question: Sorry if this is a very basic question, but I am not a physicist, but this is something that has always intrigued me. 

I have read that the properties of a black hole are its mass, spin and angular momentum. And that lower/higher the mass, higher/lower its density and so on. Although singularities are described by mathematical models, it's hard (at least for me) to believe that a singularity actually exists in the middle of space in every black hole. So why is it that singularities are actually hypothesized? Can the extremely dense nature etc. of black holes not explain its properties with a finite volume?

Also, whenever black holes are visualized in animations etc., it's always an event horizon and the observer still keeps falling through what looks like empty space. What is the significance (and the location) of the mass of the black hole itself, then, ignoring the mass it accumulates/radiates throughout its lifetime?

Answer: 
The singularity is not a physically established fact but a prediction of a classical field theory (that's what general relativity is). Such predictions are usually seen as a sign that a theory breaks down in a certain regime or on a certain scale and that it has to be replaced with a better theory.

Unfortunately for us, no measurement has been made to date that contradicts general relativity, so we simply do not even have a slight hint of evidence of what to replace it with. We can speculate about what happens near the singularity predicted by general relativity and have done so plenty, but there is, at this time, simply nothing available that could tell us which of these speculative solutions is the correct one, or if we need something completely different that nobody has guessed, yet. 

As for beliefs... that's not how science works, so there is no need to garner any. Science starts with empirical evidence and then compares it with hypothetical explanations. The explanation that fits the evidence best is eventually elevated to the rank of a theory. As of now we simply do not have a good theory of what is "inside" a black hole. "We don't know" is therefor the correct and perfectly adequate scientific answer. We think we understand what general relativity predicts to be in there... and that's called a singularity (and it's not even a point like object if I understand some of these predictions correctly). I doubt that, figuratively speaking, more than a handful of physicists think that's the entire story, though.
It might be worth looking at this in the context of a simpler theory of gravity - Newtons; there it is said that the gravitational attraction between two particles is inversely proportional to the square of their radial separation.

Thus the closer they are, the stronger the attraction; so what happens when they get infinitely close? We have a singularity ...

Singularities tend to signify that there is a breakdown of some kind in our theories of reality. 

Here it is easily fixed: particles cannot get infinitely close to each other - they are charged so at as certain point repulsion kicks in; another possibility is that extensionless particles are not possible - so strings; which resolves some of the infinities in a perturbation expansion ie Feynman diagrams.

The point then, is that somehow one has to resolve the singularity in a black hole; this will most probably be part of a full theory of QG - an ongoing project.

Both String theory and Quantum Loop Gravity, apparently have a 'microscopic' picture of black holes; this conjecturally explains black hole entropy - macroscopic datum - found by semi-classical means, by a detailed picture in the small; this then, suggestively resolves the singularity. 

But given the traditional conception of physics as an empirical science; the conjectural status of both these theories of QG - it can hardly be said to be entirely conclusive; but it is certainly suggestive.