How, exactly, does a reaction wheel work? Reaction wheels, mounted on spacecrafts and satellites, are used for precision attitude control. It is not clear to me how they can do this, though.
My best guess is that when a motor accelerates a rotating mass, it exerts a torque on the flywheel and itself. As in, if motor exerts 1 N·m of torque on the flywheel in microgravity, the system acts as though $\frac{1}{2}$ N·m was applied in different directions to both flywheel and motor.
Is this the correct way to think about how reaction wheels work?
If not, I would assume that they work by some complex interaction of the flywheels by making them precess.
 A: Reaction wheels, momentum wheels, and control moment gyros are three somewhat distinct ways of controlling the rotation and orientation of a spacecraft.
Reaction wheels are the easiest to understand, at least in their simplest form. Consider a spacecraft such as a space telescope that is nominally not rotating with respect to inertial space. The reaction wheels also nominally are not rotating. Suppose the spacecraft has just finished making one observation and now needs to rotate to some new (but once again non-rotating) orientation. The solution is simple: Set the reaction wheels in motion, against the direction the spacecraft needs to rotate. The spacecraft will begin rotating in the desired direction. The reaction wheels' rotations are stopped when the spacecraft reaches the desired orientation.
There are problems with this simplistic view. One issue is that sometimes the spacecraft needs to be rotating, such as taking a long exposure of an object in the solar system. The reaction wheels can be used to set up this rotation, but now the reaction wheels remain rotating. Another issue is external torques, and there are always external torques. Sources of these external torques include gravity gradient, atmospheric drag, solar radiation pressure, and uneven radiative cooling. The reaction wheels can be used to some extent to counter these external torques. Eventually, the reaction wheels will build up such a fast rotation that they are no longer useful. Some other mechanism, typically thrusters, needs to be used to desaturate the reaction wheels.
Momentum wheels are essentially reaction wheels in which the nominal rotation is non-zero, and typically a rather high rate. Torque is provided to the spacecraft by changing the wheels' rotation rates. The nominally high rotation rate makes the vehicle a bit immune to any external torques compared with a vehicle with (nominally non-rotating) reaction wheels, but it vastly increases the gyroscopic torques (making control more complex) and it requires more sophisticated motors than does a vehicle with reaction wheels. Momentum wheels, like reaction wheels can get saturated. They can also eventually build up too slow a rotation. Once again, some other mechanism is needed to bring the momentum wheels back to nominal conditions.
Control moment gyros are a rather different beast. Like momentum wheels, they nominally rotate at a high rate, and typically a very high rate. Gyroscopic torque is an undesired outcome in reaction wheels and momentum wheels. Unlike reaction wheels or momentum wheels, control moment gyros explicitly take advantage of gyroscopic torque. Torque is provided to the spacecraft by torquing the CMG in a direction orthogonal to the CMG's angular momentum vector. This can result in much larger torques on the spacecraft compared to those obtainable from reaction wheels or momentum wheels. The downside is that control becomes very complex, and the very high rotation rates mandates very high precision machining. CMGs are not cheap.
Smaller vehicles tend to use reaction wheels because of their simplicity and low cost. Larger vehicles tend to use momentum wheels because of their increased resistance to external torques. Only the largest of vehicles tend to use CMGs.
A: You are partially correct. If you have two objects with moment of inertia $I_1$ and $I_2$ then if one applies a torque to the other, they will start rotating in opposite directions. So if one object is the reaction wheel and the other is the satellite, the satellite will indeed rotate (while the reaction wheel, internally, is rotating in the opposite direction).
Where you are going slightly wrong in your question is with your calculation. The net angular momentum of the system will still be zero; the angular momentum of the wheel will be equal and opposite to the angular momentum of the satellite. Their rates of rotation will be in inverse proportion to their moments of inertia, so
$$I_1\omega_1 + I_2\omega_2 = const$$
And if they start out with no net rotation, then $const = 0$.
