does cold water heavier than warm water
You have to be careful what you actually mean by this question. The mass of a particular amount of water does not change due to temperature. However, the density changes.
If you take two beakers of water, each with the same number of molecules in them, then the weight will be the same. If you then alternately heat and cool the two samples, they will still weigh exactly the same (mass). However, the will have different densities (assuming they are not under pressure) and if you allow them to interact, the cooler water will sink to the bottom and the warmer water will rise. This is not because of mass, but because of density and displacement.
Interestingly, once the water gets really cold and freezes it will become less dense than water and float. This is because the water molecules are spread out differently in a frozen state than in a liquid state.