Is it possible to run Android applications on (normal) Linux?
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