OSX Mavericks doesn't reboot/shutdown/logout from menubar sudo reboot or sudo halt works when executed from terminal. But, Apple Logo -> Shutdown/Restart/Log Out User doesn't work. Strangely, Apple Logo -> Sleep works.

I've executed bunch of defaults write and setting changes in the Terminal recently, and I'm not sure which one triggered this problem. Unfortunately, I didn't take any snapshots or backed up original state. 

Which files should I look into regarding the enable/disable of reboot/shutdown/logout from menubar?

How do I fix this, or where do I start looking?
Since no one knows what you did in Terminal in the past your best option is:

Reinstall Mavericks from AppleStore.

It will not temper with you personal data.

It will however rest any changes you made to the system and preferences.

see this for instructions how to.