how would the specific heat of a metal be affected if cold water was used in the calorimeter?

Answer 1

The would be no effect unless you forgot to take into consideration that the calorimeter would also thermal exchange with the cold water until an equilibrium temperature was reached. If you left the water in the calorimeter for several minutes until there was no change in temperature before you started your lab there would be no net effect on the experimental specific heat of the metal.