1) Is it mandatory to have Windows Operating System for developing Windows 8 Mobile application by using PhoneGap or any OS(like Linux, Mac etc.) will work?
PhoneGap Build doesn't care what operating system you're using to create your app. PGB will need to know how to sign your app, so be sure to review http://docs.build.phonegap.com/en_US/signing_signing-windows.md.html#Windows%20Signing
2) Only Microsoft Visual Studio is required as IDE for development of the Mobile Application or any other IDE( Android Studio, Eclipse etc.) will work.
3) If Windows OS is not required then how to create , build, and run the Windows Mobile Application in Linux (Ubuntu).
1) Create your application code on your machine, following the format specified by PhoneGap Build. If you haven't reviewed the documentation, now is a good time to do so: http://docs.build.phonegap.com/en_US/introduction_getting_started.md.html#Getting%20Starte d%20with%20Build
2) Upload your app to PhoneGap Build. Be sure to provide your Windows signing information, but you'll only have to do this once.
3) Build your app using PGB
4) Install the app on your phone using the link/QR code provided by PGB.
If you only need access to the core plugins, you might consider using the PhoneGap Desktop & Developer apps. This lets you test your app without having to go through the build process, and also lets you quickly reload your app after changes. There are some big edge cases that you have to worry about, but I still find it very useful in my workflow.