Role of the Schottky diode (power supplies)
What is the role of the Schottky diode in this case?

If you connected the incoming supply to J5 the wrong way round, the diode (D5) would block that voltage and thus prevent U3 and U4 from burning or breaking.

How do I choose the Schottky diode? This diode is 50V / 10A. Why was a
diode with these characteristics chosen?

It has to be able to handle the full current that flows and, given that this may be several amps, a 10 amp rating may be appropriate for reliability of the device. It also has to have a reverse voltage rating greater than 24 volts and, for Schottky type diodes, a 50 volts rating is appropriate (given that they can be quite leaky).