If your dependent variable is a probability which is usually 0-100 percent, why did you use logistic regression? The fact that the data is not normally distributed is not a good reason to run logistic regression if what you are predicting ranges from 0 -100.

What do you want to test if it is right or not? The log likelihood and other model tests show you if the overall model has predictive value. The individual wald test for variables show if they have predictive value.

A practical problem of doing multiple test the way you are is you increase the chance of type 1 error but that tends to get ignored in regression models. Dropping variables and rerunning the results, even if commonly done, is really not the right way to run regression. If your theory says the variables are supposed to be in the model you should leave them there and report they are not significant.