Not much, just direct observations from several independent users. And that this should be factored in to your thinking. Potassium has been a very understudied element in aquarium water, and there is just not much info, especially laboratory study type info, about it in the context of aquaria.
However, It has been said that direct observations from several independent users can be MORE reliable
No idea, however as dosing it in tanks suffering a deficiency has produced good results in terms of colour & growth, my guess is that keeping correct levels is of benefit.
Again, I personally have no idea. Cracker, who ran experiments to determine this, does not believe it is.
Sorry I cannot provide the true scientific proof you no doubt expect me to provide, but as has been said, if your looking for true scientific proof then you will be sorely disappointed in this hobby. Often the best you will do is finding significant amounts of evidence.
That's about as full an answer as I can give at this point, to me anyway, it's a case of "it works, so I'll use it".