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I recently faced a similar situation. I would look at your data carefully -- do you have any questions where the overwhelming majority of respondents gave the same answer? A question like, "Do you have a phone?" won't be very helpful in analyzing your results because virtually everyone will give the same response. Eliminate those questions first. (In my case, it was business size -- everyone fell into the same category.)
I'm not sure a factor analysis is really what you're going for, but is certainly one way to approach the problem. If you think there is an unknown "factor" out there that is causing respondents to answer questions in a similar fashion, then I think it could be a good approach. (Eg, certain people are more likely to use up their sick leave AND they are more likely to use their cell phone during the workday -- factor analysis might reveal they are parents of school-aged children.)
Before you reject questions based on a regression analysis, have you verified your assumptions about the data? Are they normally distributed? Is multicollinearity a problem? You could be rejecting questions that might actually be helpful.
Ultimately, I addressed the question not with numbers and formulas, but with logic. Instead of worrying about predictors, I focused on "How does the answer to this question help me design my program?" You can have all the correlations in the world, but if they aren't practical, they aren't actually helpful. Look at your questions carefully and ask yourself what you learn from that question. You could discover that people who drive blue cars are more likely to buy your product; but if you can't find a way to identify blue car owners so you can market to them, the predictor isn't helpful. Focus on the questions that give you practical and usable data.
