After extracting all the files from the .zip file available at GitHub, which of the files do I need to open from the download in order to generate keys offline. Is it the bitaddress.org html file? Should I open it with my browser?
Yes, you open the HTML file in a web browser.
You can use either bitaddress.org.html or index.html
Just so you know, none of the files are signed, so you can't verify the authenticity. I believe the devs just don't give a fuck anymore :D
To be on the safe side, disconnect from the internet before you run the key-generator.