For the original case, I would expect the warmer one to reach equilibrium quicker.
This for two (probably quite small in practice) reasons. First, the warmer one will evaporate more of it's water than the cooler one. It may not be a significant amount of evaporation; but any evaporating would reduce the amount of mass to be cooled. Depending on how you set up the scenario, getting the water to 10 degrees hotter would cause evaporation before you even start the timers.
The second reason is due to thermal convection. Heat rises, so when you have water warmer than the surroundings, this would cause more airflow upwards from the container, compared to cool water where the air would be more prone to stagnate on top, reducing the convection; thus reducing the heat transfer.
Increasing surface area of the water will only make these two effects more pronounced for the warm water, by increasing the surface area for evaporation and convection.
Increasing the ambient to 80 degrees shouldn't really change this.
The rate of change will not be completely constant. For starters, as both approach equilibrium, the heating and cooling will slow down. Also, the rate may be affected by stagnation in the water itself. The cooler bucket may decrease it's heat transfer rate as a layer of warmed water begins to develop on top. This would greatly reduce the heat transfer rate on top, only getting worse as that layer gets closer to equilibrium. This doesn't happen with the warmer bucket, because the cooler room cools the water surface, and that water sinks below the warmer water, leading to natural convection.
The only way I can think of that you might get different results is if you kept the water in a sealed container, and kept it closed with an insulated lid, while having the sides of it conductive. I'm not even sure if that would work though, it would just eliminate evaporation and reduce the effects of stagnant convection.