If you copy from Word 2011 in this way, then paste it into MathType, you can then use the equation in any of 700+ applications and websites that MathType works with.
Most software applications and websites do not understand MathML, but MathType does. You mention Mathematica, and that's a good example, because using this technique you can copy from Word and paste directly into Mathematica as MathML, which Mathematica understands and can deal with. You will need to use the VL Serializer tool instead. In the Authoring and Proofing Tools section, choose Equations.
My friend who also has Microsoft Word 2011 on a MacBook Pro has his keyboard shortcuts working fine. For some reason, all the editing keyboard shortcuts - such as Cut: Cmd + X, Copy: Cmd + C, and Undo: Cmd + Z - does not work in it. You have the same options in Word 2011 for Mac you just access it in a different way, but in a way consistent with Mac OS preference settings. I have a MacBook Pro with Microsoft Word 2011 installed on it. Based on the way you phrased your question, I assume you know this already (but others may not). You can choose between MathML and 'Linear Format'. In Word for Windows, there's a preference setting where you can choose the format of the clipboard contents when you copy an OMML equation (OMML = Office Math Markup Language i.e., the 'new' equation editor).
