Is it possible to control your character's movement with the analog stick?
Yes, it is possible, via the menu brought up by pressing the Home button. Once inside this menu, do the following:
controller settings > switch analog mode > analog mode`
Setting this to analog mode will enable the analog stick. I'm uncertain what digital mode does, and the PS3 manual isn't that helpful either. The only thing I can gather is that certain games only support one of the two modes, but doesn't offer any insight on what these two modes are.