Predicted rates for solar neutrino experiments that are obtained with a modified stellar evolution code originally developed to study the advanced stages of stellar evolution are shown to be in agreement with other recently-calculated precise solar models to about 2% (0.2 SNU for the chlorine experiment). Different scenarios for pre-main sequence evolution are explored and are found to change the predicted rates for solar neutrino experiments by less than or of order 1%. The influence of the depth of the solar convective zone on the predicted solar neutrino fluxes is established by direct calculation. It is shown that a change in the calculated depth of the convective zone that is five times larger than the quoted helioseismological measurement uncertainty determined by Christensen-Dalsgaard, Gough, & Thompson causes a change in the predicted $^8$B neutrino flux of less than 7% and a change in the $^7$Be neutrino flux of less than 4%. Additionally, it is shown that the radiative opacities near the depth of the convective zone cannot differ from the standard OPAL values by more than about 12% without causing the calculated and measured depths of the convective zone to differ by more than five times the helioseismological measurement uncertainty.Comment: 28 pages. Postscript file including figure