Why 2 electrolytic capacitors and Schottky rectifier at power supply?
Why use 2 electrolytic capacitors instead of just one?


That's often a choice, maybe 2 capacitors of 470 uF are cheaper / smaller / fit better than one 1000 uF capacitor.

In high frequency switching power supplies a certain low value series resistance (ESR) is needed, that might be more easily achieved when using two capacitors


  Why no ceramic capacitors has been used? Has this been omitted intentionally?


The designer decided that these were not needed. Perhaps the low ESR could be met without using ceramic capacitors.

You might want to explain why you think that ceramic capacitors are needed!


  Why is a high value of 470uF used? I searched and found that many power supply decoupling capacitors are less than 100uF for my application where the supply is only about 12 V DC.


The designer decided that this was needed to achieve low enough ripple.

Saying "the supply is only 12 V" does not mean anything. The value of smoothing capacitors scales with the current (and switching or AC frequency), not with the voltage.


  The Schottky diode is the most intriguing to me. It is a rectifier, so is it also helping to smooth out the DC signal?


No it is simply a protection against reverse voltage. If for some reason the voltage becomes negative the diode will conduct and that way tries to limit the voltage. What happens to electrolytic capacitors which are subjected to a reversed DC voltage ? They go BANG!