Mass-density functions: how is there mass-density at points?

When we say the mass density is $\rho(x,y,z)$, we mean that the mass within any finite region $R$ is given by $$ M(R) = \int_R \rho(x,y,z)\ dx\,dy\,dz. $$ In other words, specifying the mass density $\rho(x,y,z)$ is a concise way of describing the function that takes a region $R$ as input and returns the mass $M(R)$ in that region as output.

The region $R$ can be arbitrarily small, so your intuition is on the right track. If we take $R$ to be a point, then the mass $M(R)$ is zero, no matter how large the mass density may be (as long as it's finite).

Basically, you are correct. The mass contained in a point (when we speak of continuous materials) is zero.
However, we can indeed take a small amount of length, area, or volume, mathematically described as $dx$, $dA$, or $dV$ approaching zero. These are called length-, area-, or volume elements. To find the entire mass one has to sum up all the products of all infinitely small mass densities with the length, area, or volume elements at all points in the mass in the 1-, 2-, or 3d case. This summation becomes an integral of the products of the densities $\rho$ with the three different elements (assuming $\rho$ is independent of the position in $x$, $A$, or $V$):

$$m_{tot}=\int _x\rho dx,$$

for a mass on a line,

$$m_{tot}=\int _A\rho dA,$$

for a mass on a surface, and

$$m_{tot}=\int _V\rho dV,$$

for a mass in a volume.

If the mass density is dependent on the position in the mass, just replace $\rho$ by $\rho (x)$, $\rho (A)$, and $\rho (V)$.

Substance (that makes up the mass) is discrete. We have molecules, atoms, smaller particles, etc, ...

There are hints that the space itself is discrete, too (see about the Planck length), but we don't know for sure.

Then again, sometimes (almost always, in fact) it is useful to approximate the substance as smooth and homogenous on small enough scales and use the whole calculus aparatus we have available that uses real numbers.

That's how density becomes a scalar field.