Can you calculate the properties of a substance based solely on its atomic properties? I'm trying to write some software that I can use to determine, roughly, what the physical properties of a pure substance are. I know I could just use a database of the known properties of each element, but that doesn't do much for me when I go to create molecules and combine substances. A database that lists how each element will react to another is likely impossible, so I got to thinking... there has to be a mathematical way of determining these properties!

For example, I understand that there are trends on the periodic table that show that the stronger the force between the molecules is, the higher the melting point will be. I also understand that we can calculate how strong that force is based on what we know about the atom. So why can't I find any information relating the two in some mathematical way? The same goes for color, hardness, malleability, etc. Do we really only know these values through testing? We know the values but we don't know why they are what they are?

I know this is deeply complex topic and may not even be computational possible. In those cases, I'm trying to do as much as I can to approximate.
For atomic and molecular-level properties (bond, angle, torsion energies, etc), quantum mechanics provides the equations for solving for these properties. However, there are a number of approximations that are employed, especially for larger, more complex molecules.

For bulk properties of a substance, thermodynamics and statistical mechanics provide equations for solving for a variety of material properties (critical properties, phase equilibria, various types of enthalpy, and so on). Read up on equations of state. 

There is also the option of molecular simulation. In this case you are solving relatively simple, well-established equations (eg, Newton's equations of motion in molecular dynamics simulations) repeatedly in order to compute the (usually) equilibrium properties of a substance of interest. 

In all cases, the accuracy varies, and validating results by comparing various methods is considered best practice. While there are "purists" out there who believe that all parameters in a model should be derived from first principles alone, in practice models are often fit to experimental data (or even data from quantum mechanical calculations) to improve accuracy.