OK, well first let me ask what prompted you to change the impeller the first time? Did the old impeller break apart, was it in pieces when you pulled it apart? I sure hope not. If the impeller was broken apart, and you did not retrieve all the pieces, chances are good that one of those pieces is getting lodged somewhere causing intermittent pump performance.
OK, let’s assume the original impeller was not broken in pieces. Did you use OEM water pump parts, or aftermarket? Sometimes the aftermarket stuff doesn’t fit correctly which could cause this condition. You have had the gearcase down several times, so I don’t think improper water tube/grommet alignment is an issue. Did you replace just the impeller, or did you install a complete water pump kit? If you replaced just the impeller, perhaps the SS impeller plate or impeller housing is grooved/gouged/corroded causing intermittent pump action. Is this a fresh or salt water engine? If the engine has seen salt water use, perhaps the SS impeller plate is not seating properly on the gearcase mating surface. This would cause the pump to pull air instead of water and intermittent operation. The usual symptom would be that the pump would work fine provided the engine was deep in the water running in neutral or slow speeds. At higher speeds, the water level around the pump drops causing the pump to pull air through mating surface imperfections.
I forgot to mention one simpler idea. Have you tried removing the thermostat cap and running the engine when the pump stops working? Perhaps there is some crud/corrosion/bad thermostat causing circulation problems. A bad head gasket could cause intermittent pump performance, but it would seem like you would be having running problems as well if this was the case. Nonetheless, it takes only a few minutes to do a compression test.
Finally, you mention some brown drooool coming out of the pump/exhaust outlet. Keep in mind that there is a mixture of water and exhaust that is sprayed from this outlet, so getting some drool out of this outlet when there is no water present isn’t that unusual. On the other hand, a blown head gasket might introduce exhaust goop into the cooling flow causing this symptom as well….
Sorry, have given you many different directions to go here, please stick with the simple things first, don’t go ripping the head off just yet. Feel free to post some pictures so we can get a better look at what might be going on.
Frank has posted some good things to look at concerning your starter motor. You might want to consider doing a voltage drop test on the system to help isolate problem connections/cables.