Does a constant speed prop maintain the optimal angle of attack at all time?
You may be overthinking it.  The governor will set the blade angle that provides the load on the engine that keeps it from speeding up or slowing down from the setting determined by the spring preload that resists the flyweights.  You could connect the governor to a big disc brake in place of the prop and it would work the same.
So it sets the blade AOA that gives the required load without knowing what that AOA is. It's just a spindle with weights, a spring to hold them in, a mechanism to preload the spring and an oil metering valve - there's no AOA feedback to the governor, only a speed feedback, and the input command from the pilot is just pushing on a spring.
Whatever blade AOA that is achieved is the blade AOA required to achieve the load on the engine.  Whether that blade AOA is 100% optimized to get the maximum thrust from that particular condition is another story, and blade twist, airfoil selection, blade length, number of blades, total blade area, and all that all have to be balanced against various objectives.
As to what particular aircraft speed, prop rpm and engine load gives the highest possible thrust efficiency from the propeller, you'd have to get from the prop designer and there may still be some optimization of all the parameters I mentioned toward a cruise case, or a static thrust case (like seaplane props; desirable to get on the step as quickly as possible), or a maximum speed case.