Is there a heat capacity of an object when it is melting?
Yes there is a heat capacity even at the melting point.  The reason the temperature doesn't change is that you need to add the enthalpy of melting to convert solid to liquid in equilibrium.  However, under the right conditions you can supercool the liquid under the melting point, and the temperature will behave as you would expect, needing to pull a certain amount of energy out to supercool by a degree.  But this merely delays having to deal with the enthalpy of melting until solid uncleared - it is a kinetics issue, not a thermodynamics issue.  So, it is clear that the heat capacity of a phase is continuous through the phase transition (and beyond).