One of the most influencing parameters on fault plan solutions is the depth of the earthquake and its accuracy, an error in the location of the hypocenter may change the whole nature of the solution, especially as far as the depth is concerned because this influences on the position of the polarities on the focal sphere so the fault plan and the auxiliary plan positions on the sphere are altered leading in severe cases to a completely erroneous nice looking solution. The effect of magnitude is important although very much less than the position of the hypocenter. A mechanism of a high magnitude earthquake is more trustable because 1- the earthquake is more likely to represent the true tectonic general stable regime of the fault. A low magnitude earthquake may be the result of local factors or secondary localized stress regime 2- A higher magnitude earthquake will be recorded by more stations giving more first arrival polarities and better covering the focal sphere so the nodal plans are more accurate and better correlated with the true fault plan with associated dip and slip and also the auxiliary plan. Best regards Dr Emad.

All the parameters you mentioned in your question are connected to each other directly or indirectly in most of empirical formula that calculate either the magnitude of earthquake or attenuation models.

For example 

Moment magnitude is related to the size of the rupture.

Seismic moment (Mo) = μ* rupture area * slip length

where μ is the shear modulus of the crust (approx 3x1010 N/m)

Moment magnitude (Mw) = 2/3log(Mo) - 6.06

Wells and Coppersmith (1994), have compiled source parameters to develop a series of empirical relationships among moment magnitude (M), surface rupture length, subsurface rupture length, downdip rupture width, rupture area, and maximum and average displacement per event.

As you know, most of attenuation model use hypocentral distance which is related directly to focal depth as well.

As Dr. Najib stated in his answer, focal depth is an important parameter in fault plane solution. 

Regards

 If you are using moment tensor or some other form of waveform inversion to compute the focal mechanism, the answer is no - there is no relationship other than that imposed by the physical conditions under which the earthquake occurs. if your inversion method can only use teleseismic waves, then the earthquake has to be large enough. If you are using traditional first arrival polarity methods, then depth is not the only thing you need to worry about. 

The focal mechanism of an earthquake is highly dependent on the depth of an earthquake. In fact, Moment tensor solution is one of the best ways to find error free depth of an event.

Dear Anup, please note that a SOLUTION being sensititive to the depth does not mean that the focal mechanism of the actual earthquake depends on the depth. If one uses waveform inversion to obtain a focal mechanism, one will get a solution that is independent of any prior determination of the depth and representative of in situ conditions.