I’ve done this the same way, using a DV camera to pass through the analogue from the composite connection.
I had issues capturing DV with FCPX. I used iMovie and it worked way better.
As for the speed and skipping issues, you may want to try fast-forwarding the tape, then rewinding it. Maybe since they’ve been sitting the tape is kinda stuck together.
May want to try a camera cleaner tape.
I’ve also used an Elgato USB capture device if I don’t need the raw quality. It captures directly to MP4 and seems to do its own noise clean up and deinterlace.