AC - DC Adapter Question
Yes.  Generally speaking, a DC motor spins when a DC voltage is applied to it.  It may require that the polarity of the voltage be in one direction if the motor only spins in one direction, but it will spin.  Batteries and AC/DC adapters both provide a DC voltage.

Now the specifics do matter.  If you have a motor that wants to draw a lot of current, and a AC/DC adapter that cannot provide a lot of current, then you can run into issues.