Is it possible to quit TextEdit by clicking the red x of the last open file (w/o add'l software)?
The document window's close button in TextEdit does just what it's supposed to do, it closes the document window. It's not there to close the application, that's what ⌘Q is for.

So, to answer your question, no you cannot quit TextEdit by clicking the close button of a document window by itself, you'd need to use a third party app to add that functionality.