On a constant speed propeller, how is blade pitch related to airspeed and efficiency? I'm trying to wrap my head around what the PHAK (Pilot's Handbook of Aeronautical Knowledge from the FAA) states for constant speed propellers.

First, why is it that if airspeed increases, for a fixed RPM, your blade pitch will increase, and conversely decrease if airspeed drops? Is it because it wants to maintain that low AOA (2-4 degrees according to the PHAK) with the relative airflow? If its AOA were to increase, such as not changing blade angle as airspeed drops, would drag increase causing the RPM to fall? And if airspeed is too high and AOA is too low, there's not enough drag to slow down the blades?

On takeoff, you want max thrust/power. By setting the blades to low pitch, you get max RPM because the blades take less of a bite out of the air. This also keeps blade AOA low which in turn reduces drag and allows RPM to build? Also, is max thrust in this case the result of the blades generating a lot of lift because of their high velocities from the higher RPM (versus thrust from moving larger chunks of air)?

Why is low RPM preferred in climbs/cruise? If you increase the blade pitch in these flight regimes, you get a smaller AOA with respect to wind which from my understanding so far reduces drag? But then doesn't the blade taking larger bites of air require more power anyways to maintain that RPM? Is it more efficient to give power to blades to move larger loads of air than it is to run at lower blade angles and higher RPM? 

And lastly, with regards to blade AOA and relative wind, would a blade generate more lift if it had a higher AOA, but at the expense of additional drag on the blade which in turn requires more engine power? Is this drag induced drag because it's an airfoil?
A propeller has an optimum AoA, where the thrust/drag ratio is highest. A variable prop can maintain this optimum AoA at a wide range of speeds, thereby reducing fuel consumption. That's all it does.

Fixed wing propellers have a dilemma that at take-off they require maximum thrust, and in high-speed flight the engine will need to be throttled back to prevent overspeeding. Maximum engine power cannot be utilised at every airspeed.


  why is it that if airspeed increases, for a fixed rpm, your blade pitch will increase?


To maintain optimum blade AoA. Increased airspeed causes reduction of blade AoA at constant pitch.


  On takeoff, you want max thrust/power.


You want maximum thrust. This is generated by making the propeller spin as fast as possible (not exceeding tip Mach), with a maximum blade pitch where the blade does not stall. Reducing blade pitch to increase RPM results in no increase in thrust: Power = Thrust * RPM.


  If you increase the blade pitch in these flight regimes, you get a smaller AOA with respect to wind


No a larger AoA. More thrust and more torque on the propeller axis.


  Is it more efficient to give power to blades to move larger loads of air


The engine runs most economically at the RPW where maximum torque is generated, the constant speed propeller speed control wants to maintain that RPM.


  would a blade generate more lift if it had a higher AOA, but at the expense of additional drag on the blade which in turn requires more engine power? Is this drag induced drag because it's an airfoil?


Yes and yes :)