No, if the water is moving faster, the water picks up less heat with each pass it makes through the engine but it runs through the engine more often. The water isn't going anywhere, it's being recirculated. Now if the thermostat wasn't there, it wouldn't be so clear of a picture.
And you are correct, the water pump would add only a trivial amount of heat. I don't know why I mentioned it, I just thought it might be good to.
There's nothing contradictory, you're extracting heat from the engine block which I think we can all accept as being hotter than the coolant.
More flow=better cooling, this is always true. If you're recirculating said cooling water, it equates to faster heating of the water. This is all I can say, I'm out for the night :P
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