Earthquake nucleation and early seismic propagation are studied in a two-dimensional strike-slip fault model with depth-variable properties. The fault is governed by the Dieterich-Ruina rate and state friction law. We use an efficient and rigorous numerical procedure for elastodynamic analysis of earthquake sequences on slowly loaded faults developed by Lapusta et al. <2000>. We find that for decreasing values of the characteristic slip distance of the friction law, small events appear at the transition from the locked to creeping behavior toward the bottom of the seismogenic zone. Small and large events have very similar nucleation phases in our simulations. Here, by nucleation phase we mean gradually accelerating aseismic slip in a small slowly expanding zone before the breakout of the dynamic, seismically detectable event. Moment acceleration (to which velocity seismograms are proportional) in early stages of seismic propagation exhibits irregular fluctuations, in the form of speedups and slowdowns in the moment release rate, consistently with observations as reported by Ellsworth and Beroza <1995>. Our simulations show that such irregular moment acceleration can, at least in part, be due to the heterogeneous stress distribution imprinted on the fault by the arrest of previous small events and by stress concentrations at the borders of creeping regions and to partial arrest of the rupture in velocity-strengthening fault regions which inhibit seismic slip.