I'll keep this thread to fighter planes. Why did many fighters of the same approximate power have so many different props? The Germans only fielded 3 blade props, England went all the way to 5 blades on the late Spitfires.
The US had 3 and 4 blade props. THe Hellcat had a skinny 3 blade prop, the Corsair started off with a skinny 3 blade and went to a skinny 4 blade, the P47 started off with a skinny 4 blade prop and went to a fat 4 blade prop. They all had the same basic engine with the same basic horsepower, why wouldn't they all have the same prop? Would a Hellcat have benefitted from a 4 bladed paddle prop? Would the Corsair? And we all know the story of the P38K that was never produced. WHY WHY WHY????
I hope I have pitched this at an understandable level:
As previously been stated propeller configuration depends on a great many things.
To absorb the power produced by the engine the propeller must present to the air a certain amount of surface area to move the air it has to draw through.
Firstly there is the propeller pitch which a theoretical calculation of how far the propeller with move through the air in a single rotation, this in practice is less than the theoretical calculation due to "slip" in other words inefficiencies in not all of the air being pushed back.
To a large extent slip depends on the propellers "solidity" the best way to describe solidity is if the propeller was compared to a solid disc the solidity is the difference in amount of material between the propeller blades and the solid disc. The higher the % of solidity the more efficient it becomes and there are 2 ways to increase solidity the first is to have broad chord propeller blades (favoured by German designers) or increase the number of blades (favoured by British & American designers). There is a downside of both concepts as fat broad chord propeller blades are heavy and difficult to balance and larger number of propeller blades make large bulky and heavy propeller hubs, more difficult to manufacture, more complex and more difficult to streamline thus requiring very large spinners, there is also a limit to how many blades can be fitted to a single hub, hence the concept of the contra-rotating propeller (not to be confused with the counter-rotating propeller which is driven by 2 separate engines).
The next factor is the propeller pitch the coarser the pitch then the faster the aircraft will cruise and the higher the top speed. However, coarse pitch propellers are extremely inefficient a low speed so very coarse pitch propellers impart extremely long take off runs due to the slippage previously described, hence the introduction of variable pitch propellers.
The next consideration is propeller diameter, this depends on the engine driveshaft's output speed, the longer the propeller blade, the faster the tip speed, the longer the propeller blade the lower the rotational speed which is another factor in the consideration of the number of propeller blades fitted, the propeller tip speed should never exceed Mach 1 - the speed of sound as firstly this creates a great deal of noise (a la T6 Harvard/Texan rasping noise) it also sets up shock waves and turbulence for the propeller blades following behind band this is a very large factor in the ultimate top speed of a piston engine aeroplane, where as well as the tip speed of the rotating propeller blade is added the forward speed of the aircraft, as the propeller reaches Mach 1 shock waves are produced which begin at the tip and extend down the span of the propeller blade as more of it exceeds Mach 1. the shock waves set up effectively become an air brake and prevent the aircraft from accelerating further, which can be achieved in a dive, but the vibration and forces induced will easily break the engine output shaft, a frequent occurrence on late war high speed fighters.
A very large disadvantage of increasing propeller size (and therefore the power absorption) is the torque produced by the propeller & engine combination, this at low speed can lead to a great influence and effect on the aeroplane at low speed both on take-off where a swing is caused by several forces, the torque from the engine trying to turn the airframe in the opposite direction and consequently one wheel being much heavier on the ground than the one on the opposite side, secondly, the rudder (and elevators) being largely ineffective due to a lack of airflow over them, and thirdly the spiral airflow blowing back from the propeller influencing the keel surface of the airframe (this is completely negated by a contra-rotating propeller where torque id completely cancelled out).
On approach at low speed for landing, if the throttle is opened too quickly then torque will be applied to the airframe and at low speed there may not be enough aileron & rudder authority to oppose this. This phenomenon was particularly prevalent on Russian designs with large engines and small wings, and if you study the aircraft carrier footage it will be noticed that many aircraft on approach suddenly roll over onto their backs and disappear down the side of the ship inverted, that was what we in the industry term a torque stall.
I hope this answers your question?