Do ailerons on opposite wings move together?
On most general aviation airplanes, and other cable-controlled aircraft, they move together and for that matter are cross linked to the same physical cable. You can see this nicely on the control diagram for the B-24


(source)

Some ailerons are designed to counteract adverse yaw and may move such that they are asymmetrical to the flow over the wing. 


  1) Differential Ailerons: One aileron is raised a greater distance
  than the other aileron is lowered. The extra upward aileron movement
  produces more drag change than an increase in AOA on the downward
  aileron. This produces an increase in drag on the descending wing,
  which reduces adverse yaw.
  
  2) Frise Ailerons: The aileron being raised pivots on an offset hinge.
  The leading edge of the aileron is now pushed into the airflow,
  creating drag and reducing adverse yaw. In this case, frise ailerons
  are using form drag to counter induced drag.


But not all aircraft contain such systems. Fly-by-wire systems are theoretically capable of deflecting any control surface as they see fit and may not move symmetrically but generally speaking their direction of movement is opposite. Some fly-by-wire aircraft even have split ailerons which may see parts of a give aileron deflect differently. This answer covers that quite nicely. 

If you really want to go back, the 1912 Farman HF.20 had single action ailerons that only deflected downward and were pushed back into position by the airflow. I would say (if these count as ailerons) that they do in fact not move in tandem. There were some other early designs covered here that worked a bit differently than modern ailerons. 



As for your second point, aircraft that have elevons, which serve as both the elevator and the aileron (mostly delta wing stuff like the Concorde and other high speed airframes), can move the surfaces in the same direction. Generally other surfaces are employed for the use case of needing to move both ailerons in the same direction (speed brakes).