Is it possible to run Android applications on (normal) Linux? Since Android is a forked Linux kernel plus software stack I am wondering whether there are any projects or ideas regarding creating a version of Android that can be added to a Linux system to allow a Linux system to run Android applications (compiled for, say, x86).

Does anything like that exist or would it be possible?

(I don't mean the emulator that comes with the SDK.)
Nothing like that exists yet since the apps are not run natively in the OS, but they use Dalvik Virtual Machine instead (Wikipedia). It's somewhat comparable to Java in the sense that it uses bytecode and creates a separated environment for each app. So unless the Dalvik environment is ported to other platforms, they can't be run on normal hardware.

There is a project called Alien Dalvik by Myriad that tries to run Android apps in different environments, but there's not much info on it. Myriad Alien Dalvik