Is it possible to mechanically spin a small motor to create a current, or do I need a generator? Is this true of all small electrical motors?

In general - yes, turning an electric motor by hand can indeed induce a voltage across an attached load. A motor and a generator are basically the same thing. See: Electric motors and generators (http://www.animations.physics.unsw.edu.au/jw/electricmotors.html)

Almost. There must be a magnetic field present.