if you have any such STBs in mind which I can order from China and use for my purpose?
Sorry no. All set top boxes are basically locked system that are not programmable in any way.
I dont know at the moment, whether it is even possible or not with Arduino
It is not.
There is not the processing power nor enough memory on an Arduino.
Is it possible that we design a prototype on Arduino and then take our design to custom PCB?
It is but it will cost a lot unless you are making a lot, I mean batches of 50,000 or so.