You did not keep the serial numbers to apps that cost thousands of dollars? Wow. Sucks to be you. I don't know about what version of Final Cut Pro and DVD Studio Pro you have, but the box for Final Cut Studio is not something you "lose". It's freaking huge and weighs more than the box the iMac comes in.
However, if you are legit and have registered your software (which you should), you should contact Apple for your serial numbers. At least with Logic Pro you have the hardware dongle to prove you rightfully own it. (Right...?). Take it to an Apple store and I'm sure they'll be more than happy to help you out.