"Downloading happens at the fastest possible speed, but without interfering with your Internet usage, so if you have a slow connection you'll be able to download Youtube videos and still use the Internet at optimal speed."

That can only be true if it's not using your connection (twice!) to do the download.

That's correct. It doesn't use your connection twice. It just uploads the video to your Dropbox folder after which Dropbox does the syncing.

It's the latter. Dropbox doesn't allow transloading files from arbitrary URLs, so it has to travel through my server.

Careful. That makes you liable for copyright violations. Better to use the user's computer.