For the iOS work I do I use Xcode. Unbelievably it is slightly less usable than Eclipse (this is not praising Eclipse!). I am constantly astonished at things it does. Presumably eventually Stockholm Syndrome sets in and you train yourself to not do the things that hurt. For a library I am working on that has no gui elements I am actually editing the code on my Linux box using emacs with sshfs access to the files on the mac, and a ssh terminal connection in to build/run/test from the command line. To access the Mac desktop I use Remote Desktop (VNC under the hood). One thing I praise Apple for is that their VNC server implementation is excellent - very snappy especially compared to the dismal Linux experience. Chrome Remote Desktop also works although it sometimes has problems with special keys. I have the Mac sitting in the back of a cupboard with nothing connected except a USB cable that ends up on my desk where I plugin the devices. You can do some things from the command line such as building (xcode-build). I have no idea if you can install apps, get their logs etc using command line tools.
If you don't want to write native apps then you can use PhoneGap. It is essentially a web shell (html widget and javascript) with various extra APIs (eg contacts access) and you provide your code as HTML+javascript. There is much debate about pure HTML5 webapps, solutions like PhoneGap, cross platform tools like MonoTouch versus native apps. Non-native approaches tend to be quicker to initially develop and port, but can have some rough edges and take longer to work at the "edges" of functionality.
TLDR: I have no clue about developing WP8 apps using Linux dev environment
But all in all, every phone OS (except maybe Firefox OS) puts me down on this. With iOS it's OSX and Xcode and with Android you're better to use some IDE like Eclipse.