Q: Does a USB 3.0 to Firewire 400 exist?

You should be able to transfer the files via USB .. you just won't be able to playback to through FCP via USB.  Once you've transferred the files to your computer, they'll probably need to be converted to an editable format like ProRes before you import them into your FCP project.

A more expensive way would be to purchase an SDI or HDMI I/O device for your Mac and use the old Log & Capture feature in FCP while playing back from the camera.

There's a very slight chance that Panasonic has a utility that you can download so you can use FCP's Log & Transfer function, but I doubt it's compatible with your OS (10.6.8).

-DH

There's no such thing as that type of connector.  And it wouldn't work the way you are thinking...direct import from the camera is not the way to go with tapeless.  What you need to do is offload the camera card...archive the whole thing, full card structure and all.  So USB from the camera if you use the camera as a card reader. Or, because it records to SD cards, get an SD card reader.  Then you archive the full card. Get a hard drive that you will use as your ARCHIVE DRIVE...Make a new folder, give it a unique name, and copy the entire contents of the card into that folder.  THEN you convert the files into editing files onto a different drive, your MEDIA DRIVE. 

Unfortunately you are going to have some issues with this camera and your current setup.  You are using software that was discontinued 5 years ago, and a camera that is pretty new.  There is no LOG AND TRANSFER plugin for that camera and FCP 7...so you can't use LOG AND TRANSFER.  There are a few software converters out there, but they require an OS greater than 10.6.8. And the newer OS versions don't mix with FCP 7 well. After 10.9, Compressor, Motion, QMaster no longer work. It's tricky to use current cameras and older editing software.  Might need to look at more current editing software. Or another computer to convert the footage. EditReady is good for converting this.