Just wanted to share my experience with FCP7 under my Mountain Lion install in case this helps someone else down the line.
I don't think I've done a clean install of OS X since Snow Leopard which means a "dirty" upgrade install of Lion and another of Mountain Lion on top of it. I have not reinstalled FCP7 since then either. However, for the couple of weeks since I've installed Mountain Lion, I haven't had any issues with my system running FCP7. Everything seems to work happily. I'm running on an iMac (late 2009 i7, 4G RAM).
I don't recommend not doing a clean install, but just the same, it seems to be working for me.