Water

Is water at exactly 0 centigrade a liquid water or solid ice?

It depends. Was it cooled to 0 or heated to zero? It will actually stay in its current phase until disturbed. http://www.youtube.com/results?q=water freezing  Also, pressure is a factor. http://en.wikipedia.org/wiki/Phase_diagram

>>2
Theoretically it should be in equilibrium. That is that some of the mass in the system is liquid while other is solid. However, if cooling to 0 degrees it would be liquid due to the lack of nucleation at that temp. Heating it up, it would be solid I believe

an answer to such a question will not concern the nature of science so much as it will the nature of definitions.
as such, it's bound to result in pointless hair-splitting.
regardless, physical state also depends on pressure (although i guess in your question it's implied the water is at 1atm).

Assuming standard atmospheric conditions, when a water is a liquid, it's structure is disordered.  In order to force it to order itself, additional energy must be removed.

Likewise, when ordered ice is heated to zero degrees it still does not have enough to break this ordering.  More must be added.

So, to answer your question, any mixture of normal water/ice under standard conditions will, at zero degrees, remain the same mixture of water and ice on average.  Additional energy must be added or removed to force the transition one way or the other.

>>4
haxx haxx !

Lol, this one's easy.

It's liquid prior to crystalization; post-crystalization, it's a solid.

How's that?

Simple, first year chemistry answer:

It's both (depends on pressure).

Why?

Water freezes when the temperature drops below 0C at 1atm. I say 1 atm because of reasons I'm not arsed going in to (look at LeChateliers Principle).

AT 0C, water is at an equilibrium between its solid and liquid phase, this is because there is too much energy in the surroundings entering the system causing the water molecules to gain energy and move around, increasing entropy and losing the notion of a solid state.

On the other hand, the amount of energy being put in is so little that the energy being outputted by molecular collisions (losing kinetic energy) is causing the system to decrease in internal energy, therefore the molecules are becoming less mobile, thus a solid is forming.