A family of Microsoft word processing software products for creating web, email, and print documents.
It isn't so much a Windows v. Mac issue as it is a version/file format issue.
I'm betting that the object was created in & saved as a .docx file using the newer graphics features of Office [2007, 2008, 2010, 2011]. The problem is that you're pasting to a document saved in the deprecated .doc format which doesn't support the capabilities of the new graphics engine. The shadow data is retained but can't be displayed -- that's why it shows up again when pasted back into a .docx file.
Save the file (who's title bar most likely contains the phrase [Compatibility Mode] in the Word Document (.docx) format & it should come right.
As for the doc-in-doc, yes, it travels with the host. The thing is, though, that if you save your files in the .docx format you'll be able to paste just the editable objects (diagrams) "as is" rather than having to paste in a Word Document Object. Result: Smaller, more efficient, more manageable, more stable document.