iMovie (and Final Cut Pro) are capable of controling many DV camcorders to Play, Rewind, Stop, etc. But because you're going through the Media Converter iMovie won't be able to control your camera. So you will manually have to push "Play" on your camera.
iMovie will think that your camera is correctly attached, because it is reading the Media Converter, not the camera. Just make sure you have the RCA plugs attached to the "IN" row on the Media Converter and that the button for "Analog In" is highlighted on the Media Converter. Make sure your camera is set-up in "VCR" or "VTR" mode and push "Play". You should then get a signal into your iMovie!
If that doesn't work, I would try some new cables (Perhaps the video cable is defective? It is recommended to use an appropriate cable for the Video signal which is different from standard audio RCA cables.). And, this is probably way off, but are you sure you have an NTSC (North American signal) video camera? The Sony Digital Media Converter is not capable of receiving a PAL (European/Japanese/Australian) signal and all you see is a blank black screen (although, I've gotten the audio in with a PAL video signal).
You may try to attach a regular VCR to the Media Converter and see if you get a signal correctly from there. Can you play your tapes from the camera into your VCR or TV? If you can play the camera onto your VCR or TV you should just be able to plug the cables into the Media Converter and turn on iMovie and push "Play" on your camera and then you're off and running.
I've made tons of iMovies through my Sony Digital Media Converter, it works great! Good luck, and let us know how it goes.
-Doug