I am perplexed because I have migrated from old Macs to new Macs several times, including using Time Machine backups to restore from, and I had the exact opposite experience from you -- ALL my software (including MS-Office, Photoshop, other Adobe programs and so forth) kept all the registrations and serial numbers correctly ... I too am wondering if you did a full backup/restore ...
Anyway, I think I can help you with the Office 2004 issue. You need to find certain files from the old system (e.g. from your backup) and put them into the right places on your new system. Since at least one of these is an invisible file, and I think Spotlight won't find it easily, you can download and use EasyFind
( http://www.macupdate.com/app/mac/11076/easyfind )
to search for and find these files if they aren't where I indicate below:
/Applications/Microsoft Office 2004 (or X)/Office/
OfficePID (normally an invisible file)
Users/[Username]/Library/Preferences/Microsoft
Carbon Registration Database
Office Registration Cache
Microsoft Office Settings
Get these files from your backup of the old system and put them into the same places in your new system. I THINK this will fix your Office 2004 problem. If it doesn't work, Microsoft has a support phone number to call about lost or misplaced install disks.
As for Final Cut Pro, here is where Final Cut Studio stores its serial number info, maybe there is something similar for Final Cut Pro:
Follow this path: HD >Library >Application Support> ProApps >Final Cut Studio System ID
Unfortunately, some software "hides" this registration info by putting an obscure invisible file somewhere where one might not think to look. It's to guard against piracy, but it also impedes legally registered users (like you) from doing something legitimate, like move from one computer to another or reinstall.
I think you can contact each company and explain your situation; I expect they will tell you where the file (or files) is that you need to move over. After all, you're not asking them for a new serial number.