My question is about boiling water.
   Which boils first, hot water or cold?
   The same amount of hot water, heated at the same rate, in an
   identical vessel ( metal pot)
   or the same amount of cold water, heated at the same rate (as the hot)
   in an identical vessel (to the hot water)

To make the water boil you must add enough energy to raise the
temperature to the boing point.  The temperature of cold water would
need to be raised further, so it would take more energy to boil the
cold water.  Therefore, the warm water would boil faster.

If both samples are pure water, and they were heated at the same rate,
the hot water would boil first.  This is because the water molecules
have more energy so less energy is needed for them to overcome the
attractive forces.
Although the hot water would boil first in terms of energy applied and
time elapsed, both samples would boil simultaneously in terms of the
temperature of the water and the total energy of the molecules.