Are storm windows necessary for homes in mild climates?
In my experience (midwest winters), storm windows are generally more for extra insulation purposes than severe storm protection.  

You can install exterior or interior storm windows, and they can be of varying degrees of sturdiness.  For places that do have more extreme winds, you can buy windows that are engineered to absorb a hit from flying debris.  I'm not clear on why you would wait until winter to install them in that case (post-hurricane season).

You remove the screens if there's a concern that wind can rip them out or snow drifts can tear the screen from the frame.  As @Steve Armstrong points out, you could also remove them for cosmetic reasons since they're serving no purpose with the windows closed for the winter.

In your case (and my current home in Houston), I wouldn't think either would be a necessary winterizing task.