Is there a way to get and run OUYA's operating system/environment/whatever you want to call it on the desktop? By that I primarily mean within a simulator of any sort (e.g. Android's variant of qemu would be okay too).
Is the OUYA software even openly available for download?
The Ouya ODK is available and free. You should follow the Ouya developer documents for a walk-through of setting up your Ouya development environment. It tells you where to find and configure everything.
Once you have your development environment set up, you can create a Android virtual device (AVD) that closely resembles the Ouya. This procedure is also described in the aforementioned document. (It is also trivial test your project on an actual Ouya device, if you have one handy.)