Is it possible to quit TextEdit by clicking the red x of the last open file (w/o add'l software)? If you open TextEdit and then click on the red x in the top bar of the document, TextEdit will remain open. This is evidenced by the little black dot under TextEdit, despite no TextEdit documents being open. Why Apple designed it like this? I haven't the slightest idea.

I've found a piece of third-party software that offers this functionality called "RedQuits."

Is it possible for TextEdit to automatically quit itself when the x is pressed, without installing additional 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.