There is certainly no reason this shouldn't work, if you've got a good connection.
Does your computer register the FireWire connection? Is there a little camcorder icon on your taskbar? Are you able to cpature with Windows MovieMaker? If so, then that limits the problem to Premiere Elements.
Is this the only camcorder you capture video from? Have you gone to Premiere Elements Device Controls and ensured it is set up for DV capture?
Worst case, there are free programs, like WinDV, that will capture video in a format that's compatible with Premiere Elements.
I am concerned though about your inability to edit your captured footage. Do you have the latest version of Quicktime and the latest drivers (from the ATI or nVidia site) for your graphics card?
I have a similar Sony camcorder.
With mine, it has to be plugged into the recharger for the iLink to work - you can't just run the camcorder on battery power (even though you may have plenty left).
Also, the button that controls the camcorder function (record | play/edit | camera) has to be set to play/edit. I had once accidentally changed the mode to camera and it wouldn't recognize it.
I do not know that camera, but with many, there is a set scheme for the connections, to get things to work. It seems to differ by make/model, but many have found that just by changing the order of turning things on, and invoking the Capture module, things will usually work.
The connections are camera to computer via FW cable, then turn ON camera in proper mode, then launch PrE, then go to Capture. Users have found that their camera will require, say, plug-in, launch PrE, THEN turn ON camera, then launch the Capture module. As you can see, there can be variations in when one does each part of that chain, and it can make a difference. I would sit down with a pad and pencil, and test each possibility. If one works, that would be the scheme that I would then use.
PS - the only thing that I would NOT do would be to turn ON the camera, prior to connecting via the FW cable. That can cause big issues.