I assume that the Thrust Available/Required graphs behave in similar fashion?

To some degree. If you plot drag in INDICATED airspeeds, it's the same at every altitude, whereas the power curve shift upward (no tilt with IAS). Thrust Available will reduce with altitude. I haven't come up with an analytical demonstration of why Vx would increase with altitude, though, even though I know it has to do with the propeller/engine, rather than the aerodynamics of the airplane itself.

For most graphs showing Thrust Available/Required it shows that Thrust Available __DROPS__ off rather dramatically as higher regimes of flight are encountered. This makes sense considering a fixed pitch prop but doesn't seem to hold true for a constant speed prop where you should get max efficiency and thrust through all regimes of flight. Yes? No?

Unfortunately, no. Thrust drops off sharply for any propeller airplane, even for turbo fans. The only time you might see constant thrust with airspeed is a pure turbojet.

Remember that Power = Thrust X Velocity, so you if you have a relatively straight POWER available curve, such as with a CS prop, you can't have a straight thrust available curve. The math just doesn't work.

The way I think about it is that a propeller can only accelerate the airflow to a certain speed; as the aircraft flight speed increases, the increment in velocity the prop gives to the air in excess of the flight speed gets smaller and smaller, meaning less thrust. At full cruise speed, the prop may be imparting maybe 10-15 knots of velocity over what you're flying.