An atomic species defined by its number of protons (usually denoted $Z$) and its number of neutrons (usually denoted $N$) is called a nuclide. For atomic species the number of electrons is the same as the number of protons (i.e. $Z$). You are right to assume that the nuclide of a single nuclide solid will typically determine its melting point and hardness (although the number of neutrons plays a marginal role and some species exhibit allotropy). There is, however, no simple relationship.
From quantum mechanics we can derive a number of physical and chemical properties of a nuclide (mostly dependent on $Z$), but to answer the question of melting point we also need solid-state physics and thermodynamics/statistical mechanics. Even though we may have physical theories that explain how $Z$ relate to the melting point, we may not be able to make accurate, purely theoretical predictions of the melting points based on $Z$ alone. Predicting crystalline structure, for example, is certainly a non-trivial problem.
The prospect of creating synthetic nuclides with properties superior to those of the natural occurring ones has allured scientist for a long time. One problem is that synthetic nuclides tend to be unstable, which is typically the reason that they are not naturally occurring. There are, however, theories that predict an island of stability for certain super-heavy nuclides. Synthesizing these, however, have proved to be very difficult, so the existence of this island remains an open question.