If you add cooler water to hotter water is the end temperature the average between the two?

It depends on how accurate you want to get. For most purposes, saying its 75 degrees would be close enough, but its not completely accurate. To be more accurate, you would need to add equal masses of water together, not volumes (since the density of water changes with the temperature). Even then, it's still incorrect because the heat capacity (amount of heat energy it takes to raise its temperature) also varies by temperature. Generally, the higher the temperature, the higher its heat capacity. Or, for example, it actually takes slightly more energy to raise water from 86 C to 87 C than it did to raise it from 85 C to 86 C. What this ultimately means is that mixing 1kg of 100 C water and 1kg of 50 C water will end up with 2kg of slightly above 75 C water; eyeballing it off of a chart puts it at about 75.15 C. Unfortunately I don't have the equation for the heat capacity of water in front of me to give you an exact answer.

As long as it's the same volume of pure water, then yes.