How did micrometeorite flux change with the age of the solar system? Do we also have functions for the micrometeorite flux on the Moon, Mars, and even any random body in space as a function of solar distance too?
 A: The short answer to both your question in the subject line (change with solar system age) and your topic (function of solar distance) is "no."  We don't even know the impact flux of asteroids and comets as a function of solar distance nor time.
The work that most of us in the field cite whenever discussing crater chronology (using number of craters to determine age) is from 2001 (Neukum et al.).  The Neukum chronology does indicate a change in the impact rate through time, and while we all use it, we also (almost) all know it's wrong.  It doesn't take into account any spikes that likely occurred when asteroid families were formed, it doesn't take into account the late heavy bombardment, and it doesn't take into account other factors.  It's also formed by fitting data to basins on the moon from returned Apollo and Luna samples, but those are also disputed, especially the Copernicus crater age.  So, since we don't even know the flux of objects that form kilometer-scale impacts, we certainly don't know the change with time of micrometeorites.
In terms of solar distance, this is something that we know even less well.  Yes, you can project from the moon to other bodies, as we do with Mars (Hartmann, 2005, for example), but that doesn't take into account things like a difference in timing for the late heavy bombardment -- it really only takes into account relative proximity to the asteroid belt and velocity of impactors.  Once you go outside the inner solar system, it's completely up in the air, though people still try to do chronology based on what we know from the Moon.  The problem with going outside the main asteroid belt is that the relative numbers of asteroids to comets as impactors may change dramatically ... we just don't know.  So again, since we don't even know the impact rates of kilometer-sized objects very well, micrometeorites are again an unknown.
