Firefox 29 and later have a new Sync version that use a Firefox account and only use the name and password and no longer use a separate Sync (recovery) key (the Sync key is automatically derived from the password of the Firefox account).

The new Sync version is completely different from the version that was previously used and you need to install a Firefox 29 or later version on all connected devices.
You can enter the name and password of the Mozilla account on other devices that have a Firefox 29 or later version to add extra devices and you now longer can use a pairing key to pair a new device although this might be possible again in a future version of Sync if they decide to re-add this feature.

• https://support.mozilla.org/kb/how-to-update-to-the-new-firefox-sync
• https://wiki.mozilla.org/Identity/Firefox-Accounts

Then how am I supposed to recover my old profile?

My hard drive failed yesterday so I have no record of the old account on my PC currently. Forgive me for being a little annoyed, but isn't the point of backing up files to avoid having THIS PROBLEM I AM HAVING NOW?

Is there a source I can enter my recovery key and download my old profile and transfer everything from that to my new one with email and password?

You would have to install an older Firefox version temporarily and connect to the old Sync account and then update again to the current release.

Firefox 28.0 for Windows:

• http://download.cdn.mozilla.net/pub/mozilla.org/firefox/releases/28.0/win32/en-US/

Thank you, I appreciate that.

Although it restored everything but my bookmarks and tabs (which are the most important things)

Is there something I am missing here? I know I had everything ticked as in tabs and history and bookmarks, etc.