How can you download anything without an OS etc? If you have two drives and your OS is on one and the one you store files on is broken then using an external drive to download the game should work I think. My priority in this case would be to sort out my PC problem first!
I think what you're saying is the DVD Rom drive doesn't work or is not present. If that, then yes the download copy should work fine, and if it's just a file you download then install then yes you can save it to a usb stick. If not, it should be available to you from Amazon if you ever need to download it again to reinstall or something.
Because of FFXIV's nature I think you can copy the contents of the dvd onto another device and still run the install from that device. Another thing you could do is share the dvd drive on the laptop and map it to the machine that doesn't have a drive. Worth a shot anyway. The game doesn't require the disk to run once it's installed.