Final Cut Pro 7 will install under Mountain Lion and should run without real issues. You can use your original serial number.
If your employer purchased the computer and software, then it is up to them to insure it. If you purchased it then it is your responsibility to insure it. Otherwise, it is uninsured.
As for software the serial number is on the box in which it came unless you downloaded it from the App Store. In this case it is tied to the Apple ID that purchased it and can be re-downloaded free of charge.