DIODE heating of junctions
The PN junctions in a diode are so small that it would be difficult for any temperature gradient to exist. Also realize that a silicon PN is on one side 99.9999% silicon and something else and the other side is 99.9999% silicon and something else as well. So basically its mainly silicon. The P and N doping does not change the properties of the silicon that much.

For a schottky diode there would only be a difference if the heat resistance of the metal and silicon are very different which I doubt they are. Silicon is also a metal !

How is common Cathode more common ? This might have nothing to do with the properties of the diodes but more with the way that they are used in circuits.