Not really. There is no OS right, but there is a chip that comes with a firmware(assuming a very limited OS).
Nope! The game is just a compiled, ready to run codebase, that could communicate with underlying(console) API/ABI for good.
If you note, there are some emulators for legacy systems, like NES, so you could play NES ROM games on a PC for good. That emulator, just emulate that legacy some MHz process, with a GHz CPU which is not so hard of course!
That's not really simple, it may be complex(different units to functional sync), but overall for NES system for example, it won't be so hard. but time-consuming if you like to do it right!
I suggest you start with a very well-known platform like AVR, or PIC for start(while ARM Cortex M0 is a good option too). You have to code the firmware first(the code that runs the console). It has to support API/ABI for all lower-level peripherals(such as display, audio, IOs, etc...) and that end, the game programmer, will utilize your API/ABI for making the game!
Very great journey!, but make sure you are hardworking and passionate too much, otherwise you will give up.
Happy programming