I made a small trial using Sync Handheld. Only the video file was synced to the USB drive. My theory about the cover art file is it is a separate file in the same folder as the video file. With my music files, cover art is included in the file tag. I don't know why the sidecar wasn't synced.
Comment: it was funny how long Sync Handheld took to read the USB drive and get ready to sync. There were something like 15k files on it.
I also did a small trial with a couple of movies. I did a "Windows drag-and-drop" of two movies, sidecars and cover art files. Then I <imported> the files into a Library. Finally, I used Rename, Move, Copy to reorganize the files into my standard file structure. That seemed to work.
A couple of questions: when a sidecare file is created, is it always given the same name as the video file? Similarly, if Get Cover Art is used, is the .jpg file also always given the same name as the video file? If both of those answers are yes, I think I can used that drag-and-drop then R-M-C method efficiently.