I have FCP7 and FCPX installed side by side. If you have Final Cut Studio installed and then install X from the App Store, it will move your legacy apps into a “Final Cut Studio” folder within Applications.
Make sure your Mac is qualified to run Mavericks, and the more RAM you can throw at it the better. I would advise making a complete clone of your internal drive (if not two) as a safety precaution, then wipe the drive and do a clean install of Mavericks, FCP7, and FCPX in that order. It’s extra work, but I think you’ll have a more stable system with a clean slate.
I have switched between FCP7 and FCPX regularly in an edit session, primarily when I need to master to HDCAM.
Hope this helps,
Bill