Most of the download problems seems to be fixed by just deleting the temporary downloads folder and letting it redownload whatever it was trying to get.

To do this, go into your Documents folder (wherever that is) then "My Games" and then "FINAL FANTASY XIV - A Realm Reborn" and finally "Downloads".

From here, delete the ffxivpatch directory (it'll be recreated when you run the launcher again). Don't delete the ffxivupdater.exe file, just the ffxivpatch directory.

And then just rerun the launcher and hopefully that'll fix it.