If I've previously bought a song in iTunes, can I download it again on another computer without paying for it a second time? Both computers are authorized to the same account.
No, that is not possible. That is described on that apple support page.
You will be able to copy it yourself, with a USB stick etc, or if you are running the latest version of iTunes and both machines are on the same network you will be able to use Home Sharing to identify which tracks are in one library but not the other, then copy them across. You can even set Home Sharing to automatically sync new iTunes purchases.