Is FCP 7 compatible with 10.8 OS?
What kind of trouble has anyone encounter with FCP 7 after update to Mt. Lion? Are there any Apple patches? (have found none)
Apple stopped developing fcp7 a few years ago, well before Mt Lion was released so don't expect any updates from Apple. As far as problems go, I've had it installed for about a month without any issues. Do some searching here for other peoples experiences. FCP is an extremely complex program so problems that people have may or may not have anything to do with Mountain Lion.
To avoid any potential problems, it's always a good idea to follow accepted best practices
https://discussions.apple.com/docs/DOC-2861
And it's always a good idea to have a clone of your startup drive before any major upgrades so you can return your system to it's working state.
https://discussions.apple.com/docs/DOC-2494
If you've got a macpro tower, you can have multiple drives installed internally so having multiple OS startup options can be a good idea. If you're working with a mac that doesn't allow multiple internal drives, you can always partition your internal drive and have mutliple startup options.