The E6600 is the socket 775 desktop variant, while the mini uses the socket M laptop chips. You'll need to find a socket M processor and the model numbers are usually preceded by a T rather than E, such as T6600.
Nope, that's a Conroe (desktop version of Core 2 Duo) - the mini uses mobile versions of the Core Duo (Yonah) processor. They are socketed and it is possible to upgrade them to Merom (mobile Core 2 Duo) processors but not Conroe processors.
Apple isn't going post what exact CPU is in there because they don't want you taking it apart. I would make sure that the current cooling system will support a 2.33 GHz C2D before doing it. I'm pretty sure it will fit in there. Someone has had to of already tried this.