For example: is the hash enough to "prove" you have a file? Or do you need the file size (and potentially other properties, such as the first block) as well?

The only way to find this out would be to look at the protocol that the proprietary client uses.

They also had a service that let you predownload files from http/ftp servers and then you could request an offline broadcast from their servers to your home pc.

Instead of refetching every file again and again, they also optimized by only checking the file size + file name. So, someone came along and created a fake ftp server script, which just replied with a listing of the things you wanted to download, and they there instantly added to your account. You only had to know the filename + filesize.