does the temperature increase during melting?

Answer 1

No. (In theory)
During the process of melting, the temperature of the ice remains constant. Water freezes at 32 Deg. F. (0 deg. C.)
It melts at the same temperature.
Heat has to be added to ice to melt it, but the temperature won't increase until after the ice is melted. Then added heat can raise the temperature of the water.
The same principle applies when freezing water. The temperature will remain constant until the water is completely frozen. Once frozen, continuing to remove heat can lower the temperature below the freezing point.
The temperature at which water freezes (or melts) will vary if the pressure (such as air pressure acting on the water) is changed.
Another factor (and the reason I said "in theory") is heat variations created by convection and conduction.
Example: if water is placed in an enclosure where the air temperature is very cold, say minus 40 deg., the frozen portion (the ice) may be colder than the remaining liquid water during the freezing process due to the fact that the water can circulate and maintain a more constant temperature due to convection. The frozen portion may continue to get colder because it is now in the solid state and the temperature change created by conduction progresses at a slower rate. This could allow the portion of the ice that is in contact with the very cold air to further drop in temperature prior to the moment when all of the water has converted to the solid form.

Answer 2

yes