Buck Thorn has already addressed your second question, so allow me to address your first:
Since the temperature doesn’t change, all the thermal energy is used to increase the potential energy(is this correct?).
Not in this case.
Temperature is not a measure of kinetic energy. It is, more precisely, a measure of the kinetic energy per available degree of freedom. See my answer here: What exactly is temperature?
Intuitively, liquid water has more available degrees of freedom (rotational and translational) than ice. Thus, at the same temperature, liquid water will have more kinetic energy than solid water.
Think of it this way: As we change solid water into liquid water, we need to flow more thermal energy into it to "fill up" those additional kinetic degrees of freedom, to keep it at the same temperature. Consequently, the thermal energy used to melt ice increases both its potential energy and its kinetic energy.