FCP 6 is not. It won't even install. The installer is PowerPC based, and Mountain Lion doesn't have Rosetta...that allows PowerPC apps to work. You might be able to CUSTOM install Rosetta from Snow Leopard. But FCP 6 is VERY old software, and old software and brand spankin' new Operating Systems don't generally mix well. (FCP 6 is 6 years old, ancient in software world)
FCP 6 might work. As for Avid...only Media Composer 6.0.3 and 6.5 are "compatible." Meaning those are the ones that are tested and certified solid by Avid. Other versions aren't rated to work with ML...so you might have issues.