Is it okay to take off the propeller to run an outboard boat motor in a bucket of water?

yes, you can, it has an internal water pump. However, do not over rev it. Keep the RPMs low or you could blow the engine. Make sure the lower unit is submerged in water. depending on your engine, you will see a small stream of water exiting the top of the lower unit (either right away, or after it gets up to temp).

water pump does not depend on the prop at all. like the other guy said, don't rev it up with no load -- very bad for the engine.

Go and buy yourself a set of "ear muffs" that connect to the garden hoses, and slip pover the water intake at the bottom if your outboards leg. turn the hose on and start it up

You should always remove the prop when running the motor in a bucket or on muffs, safety reasons. So yes it is ok. Remember to lubricate the prop shaft with the proper lube when reinstalling the prop.

yes its okay, but certainly not needed.the water pump works in neutral gear the water intake is either under the cavitation plate or on either side of the gear casing just above the prop. and the water pump will draw the water in by the action of the impeller the intake must be submerged in the water . and make sure the engine is in neutral gear. also if you use "ear muffs" ie (they clamp around the gear casing for the side intake engines) you must see that they seal properly and the water is spurting out the telltail (small hole at the top of the leg). but there is no need to remove the prop. because you should flush the engine every time you use it in salt water