I spot two pairs of related quantities in your list.
Speed (I assume you mean orbital speed, not how fast the planet is spinning about its own axis) and distance. The orbital period and the semi-major axis of the orbit are related by Kepler's third law. For a circular orbit, the semi-major axis is the radius (distance) and the period is trivially related to the speed ($v=\frac{2 \pi a}{P}$, where $P$ is the period and $a$ is the semi-major axis). For elliptical orbits the speed changes over the course of an orbit, so the period instead tells you something about the average speed of the planet. Planets with large semi-major axes might move faster than planets with smaller semi-major axes at a particular instant (depending on the eccentricity of the orbits), but averaged over an orbit larger $a$ always means lower average $v$.
Mass and size are also somewhat related, but this is something of a complex topic. More mass means stronger self-gravitation, which tends to compress the size of the planet, but it also means more matter, which tends to grow the size. A planet of a given mass will have a size such that the inward pull of gravity is balanced by the outward pressure supplied by the material making up the planet. For a gas giant/star this is called hydrostatic equilibrium. I'm not sure if there's a similar term for a solid/molten planet but things are complicated somewhat by these condensed phases of matter. If you're trying to come up with a simple model, try picking a mass at random, and look up typical average densities for solid and gas planets. Pick a type of planet then use the average density and the mass to compute the radius.
There is no strong evidence that supports smaller planets typically being closer to their stars than larger planets (which is the case in our solar system); many stars have been seen with gas giants orbiting as close or closer to their star as Mercury is to the Sun. It is generally believed that these gas giants formed further out and migrated inward.
The rotation speed of a planet (about its own axis) is initially fixed by the total angular momentum of the material that collapsed to form the planet. Over time, however, tidal locking can influence the spin of two orbiting bodies. This is why Earth's Moon has period of revolution equal to its orbital period, for instance.
A planet/moon that gets too close to another body with strong tidal forces can be ripped apart.
There can also be favoured/disfavoured distances for orbits due to orbital resonance, or a planet can be entirely ejected from its orbit.
The interaction of a planet and its moon(s) is essentially the same as a star and planet system, unless you expect effects particular to a star to be important (one example would be radiation pressure).
In the end, to get a more realistic simulation going, I'd worry less about initial conditions and more about making sure your method is working accurately and accounting for not only gravitation between the star and the planets but also gravitation between planets and tidal effects as well. If you achieve that, instabilities will make sure that any "unrealistic" scenario will decay into a stable configuration over time (potentially ejecting one or several planets from the system in the process).
