For covalent compounds, what is the microscopic difference between a solid and a liquid?
Let's break this down, and keeping it to simple covalent compounds...


  Are different forces dominant in between solid and liquid molecules? By "forces" I mean ionic bonds, dipole-dipole interactions, London forces, etcetera.


All those forces (if applicable) will still exert their expected effect and there is no 'dominant force' which increases in strength when temperature rises. Just that when we raise the temperature, the molecule may have enough kinetic energy to overcome London Dispersion but not ionic forces, etc.


  What is the difference in the geometric structure of the molecules' interactions? Solids are fixed and gases are mobile, but what precisely are liquids? Solid structures with enough degrees of freedom to pivot fluidly? Do liquids generally have the same structure (simple cubic, hexagonal, etcetera) as their solid counterparts when they do interact?


So let's take a look at a typical phase transition diagram, that of water:



At the flat stages, water is changing discreetly and exists either as solid (rigid lattice) or liquid (completely formless and structure-less). And if you look at ice melting, that is exactly what you see and experience.

But the source of this question is probably the fact that you've also seen certain solids soften under heat without melting completely. These covalent materials are called thermoplastic and have a property called the "Glass Transition Temperature". The Stiffness-vs Temperature graph looks like this:



In this state, the melting solid is able to overcome the forces that hold it to a lattice, but is not able to overcome the forces that hold the molecules together and so the molecules cannot 'flow'. The reason this exists for some compounds and not others is largely a property of how their molecules hold each other together.