If water is at 0 degrees C, and ice is added, what will happen to the mixture?

if the ice is at 0 'C too, nothing at all

Water has a high latent heat capacity as well as high energy requirements meaning much energy is needed to raise its volume temperature by 1C, so by adding ice the temperature does not decrease and the ice simply melts from different temperatures. I think if youre meaning in terms of whether the volume of water completely turns to ice at this low temperature, I think it highly depends on the amount of ice, volume of water and environmental factors. Hope I helped :)

When it is at 0 degrees, it is already in ice state. Therefore, when ice is happened, nothing happens.
Ice will just fall on the the mixture with it. What you will see is a pile of ice. Simple enough.

well nothing. if ice will be a little colder that 0, that overall temperature can be a little decreased. It's depends on the amount of water and ice.

...
assuming that the ice is also at 0°C
and that the surroundings are also at 0°C
then the relative amounts of ice and water will remain constant over time
however
melting and freezing will continue at equal rates
so if you added, let's say, an ice cube
then over time the total ice will get randomized into a floating slurry.
but the amounts of each, ice and water, will stay the same.

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That is called a triple point because the water can be solid, liquid, and gas all at the same time. It takes heat to melt ice, so there is no temperature change going from ice to liquid water at 0 C. Not much evaporation happens, but there is some and it leads to what's called "freezer burn". Freeze lemon juice in cubes and then bag the cubes. They will turn into a bag of yellow mush as the water evaporates one place and condenses another.