I'm on Xamarin Studio 6.0.2 (Build 73), running on Windows. Even when I create a fresh new project, the Project Options menu entry is disabled, which means I have to edit the .csproj file in Notepad++, which is not very convenient.
Some other menu entries are also greyed out. As I said, this happens even for New projects created by Xamarin Studio.
So having looked at your version information it seems like Xamarin.Android is not installed which is why you are having this issue.
You will need to install the Xamarin.VisualStudio package. Even if you only intend to use Xamarin Studio, this is required as it includes both Xamarin.Android and Xamarin.iOS.
Please reach out to contact@xamarin.com and we can provide you with the link for this! Please add the link to this question in your email.