Is there a way to move Internet Explorer 11's tabs BELOW all tool bars, directly above the webpage, as in Firefox?
No, this is not possible as of Internet Explorer 9.
The best that you can do is to make the tabs appear below the address bar, which you can do by right clicking the title bar and selecting "Show tabs on a separate row", but this will still show the tab bar above the toolbars.
Or, another alternative is to downgrade to Internet Explorer 8, which by default shows the tab bar below all toolbars and is the default for Windows 7 (unless you own an OEM edition and the OEM chose to preinstall a newer version, but you can still downgrade). You can do this by opening Windows Update, going to "View update history", clicking "Installed updates", then uninstalling the Internet Explorer 11 update to downgrade to the last version of Internet Explorer you installed on the machine. Then, uninstall older versions until you have the original installation of Internet Explorer 8.
I accept that I am replying to an old post, but...
You probably don't want to downgrade to IE8 just for being able to position the tabs where you want. IE8 is an old and now very slow browser - according to benchmarks, all the current common browsers are faster in all ways than IE8.
IE11 might not give you the visual customizations you want, but benchmarks all show that the current version of IE11 is now faster than all the equivalent common browsers for PCs. (Chrome & Firefox). Though, if you ever thought to use Chrome purely for speed - you're now slowing yourself down over other browsers.
In essence, do your homework on browsers before changing, or just stick to IE11 and put up with the minor inconvenience regarding tabs positions. (Ctrl + Tab lets you switch to the next tab anyway)