Opposite of IE Tab / Mozilla rendering for IE
Solution 1:
Why not use the Google Chrome Frame? It's a plugin that makes the page use the WebKit rendering engine and V8 javascript engine.
Chances are if you have something that works in Firefox it will work with WebKit. And IE users are used to installing plugins for any given page so it's not much of an inconvenience to them.
Solution 2:
MSIE lets itself be embedded in other applications (because it's an OLE application). That is why Firefox can launch IE inside a Firefox tab. This seems to work so easily because Window always has IE installed.
The opposite is not possible; IE can't launch Firefox inside an IE tab because Firefox itself it not made to be embeddable in other applications. Even if embedding were possible, you would still need to have Firefox installed (you mentioned that you wanted to avoid that). You can't get the Mozilla rendering engine without installing it...
This is why any Windows application that needs a browser, either uses IE or contains it's own little browser application.
There was, once upon a time, an IE bookmarklet that would open the current webpage in Firefox - but that still requires Firefox to be installed.
Solution 3:
You can try Lunascape:
Lunascape is the world's first and only triple engine browser.
You'll have the choice between three browser engines: Gecko (Firefox), Webkit (Safari, Chrome) and Trident (Internet Explorer), now you can even run all 3 engines side-by-side in a single web browser.
Lunascape also supports a wide array of Firefox extensions.