Legal? Rooting an Android device and installing something like Cyanogenmod is legal; the Supreme Court says so. Apple tried to stop people from jailbreaking iPhones (the equivalent of Rooting Android) and lost.
Authorized by Google? Android is Open Source. Look into what that really means, and realize that Google is merely the biggest developer. Unless Google wants to violate GPL licensing terms, Google’s approval is irrelevant. That said, Google tends to be agreeable anyways. For instance, while they won’t allow third-party Android distros to include Google apps (Maps, Play…), they will (and do) allow them to distribute those apps in a separate package. When I installed Cyanogenmod on my Nook Color, I merely had to download/install 2 packages (the Android OS and Google Apps) instead of one.
So the only real problem is finding a version that is fully supported by your device. In general, such upgrades come in two flavors; Stable, which have been proven to work, and “Unsupported”, which include works-in-progress, similar-but-not-the-same versions that work on nearly-identical device, and just outright kludges. I run a Stable release of Cyanogenmod on my NC and all is well. But that’s on their list of Supported Devices, so a Stable release actually exists; I’m not sure if one exists for your device, and I really wouldn’t try the other kind.