In order to define something hot or cold, a reference temperature is essential. Often the ambient temperature is considered to be a reference temperature and any object above this temperature is defined hot and cold below this temperature. 

At ambient condition (Tamb = 25oC, Pamb =1 atm pressure), ice would appear cold, since it is at 0oC (ice melts at 0oC and water freezes at 0oC), well below the given reference temperature.

Now, cold does not necessarily mean absence of heat. If you put an ice cube (0oC) in liquid N2 (boiling point : -193oC), the latter would start boiling/vaporizing instantly because of the huge temperature difference. So, ice is cold but is hotter compared to liquid N2.

When an ice cube is placed in a water bath, it quickly draws the latent heat of melting/fusion (336 KJ/kg) from the surrounding water which is at higher temperature than it. The direction of heat flow is always from higher to lower temperature body. As a result, ice gains heat and undergoes a phase change from solid  to liquid state. Surrounding water bath looses heat and gets cooled with an apparent reduction in temperature (sensible heat transfer).

Now, this is an unsteady state heat transfer process wherein temperature of water bath decreases first and then rises again after some time due to heat transfer (natural convection)  from the surrounding air. The heat transfer mode here again depends on the temperature difference of the two bodies.

In a typical winter condition, surrounding air temperature is low, so the water bath stays cool longer due to less heat transfer from the surrounding air. But in a summer condition, due to higher heat transfer from surrounding air, the water bath will not remain cool longer and would reach a thermal equilibrium with surrounding relatively faster.

One of the laws of thermodynamics states that the rate of change of temperature is proportional to the temperature difference between the two media. So when a glass of water at room temperature has ice cube in it, at first the cube absorbs heat from the water and melts rapidly while bringing down the water temperature. When the cube has melted completely, there is equilibrium and the water is at a lower temperature than before but not as cold as the ice cube. Now the rate at which it warms up to reach equilibrium with the room temperature will depend on how much difference there is but will be slower than ice and water.

As the previous answer above mentions, cold and heat are relative. If you asked liquid nitrogen it would say an ice cube is boiling hot!