Am I correct in saying that if my results from multiple logistic regression show that the full model does not differ significantly from the "constant only" model (i.e P > 0.05 in LR goodness of fit test), then I should not read too much into individual predictor variables which have statistically significant coefficients (i.e. P < 0.05 for P>z)?

Also, if I want to control for the effect a confounding factor (e.g. gender) on a binary response variable (e.g. reads Time magazine yes/no) in relation to the predictor variable of interest (e.g. level of education), I assume I do this by including both gender and education as predictor variables. However, if the p value for gender is not significant when I run the model, but is for education, can I assume gender has no effect on the relationship between education and whether or not a person reads Time magazine, and therefore drop it from the model?

Thanks