Q:

How do programs shutdown windows?

Upon doing some looking through my computer, I found a program called shutdown.exe and another relevant one which I forgot (shutdown.exe.mui ?).
This brought forward a question. 
How does a program shutdown? 
How does it bypass shutdown -a? Would a program-issued shutdown not work if shutdown.exe is missing? If so, does this apply to the shutdown button on the start menu?

A:

How does a program shutdown?

It calls the Win32 API ExitWindowsEx() function, which does some Windows-version-dependent magic to ask the system components to turn off the computer. JdeBP has an article about what happens next (on Windows Vista/7/8).
(There are also a few other shutdown functions.)

How does it bypass shutdown -a?

By not giving you any time to run shutdown -a in the first place. The shutdown begins immediately. (shutdown -a cannot abort a shutdown in progress; it can only cancel a scheduled one.)

Would a program issued shutdown won't work if shutdown.exe is missing?

Yes. Practically all programs call the Win32 API functions directly.
In fact, programs wouldn't have worked for many users if they did use shutdown.exe.

