1. Increasing CL affects your FICO score only to the extent it effects your utilization percentage. There is no weight given to credit limits per se. For FICO, it is just your utlization.
2. Based on the fact that you are saying you have no debt, it appears it would not affect your utilization, so you would not see any FICO increase to offset the hit from the hard pull. You should expect to lose a few points for the hard pull.
3. Probably doesn't matter when you do it. Do you have any large purchases, home or auto, coming up? Do you have a lot of other inquiries?
Starting Score: ~500 (12/01/2008)