SOLUTION:
This is not a new issue, and yes it ***** that Mac will just simply post the answer here as its extremely easy.
The problem when you get that "The document Untitled could not be saved as "-----" . The file doesn't exist" message is actually nothing more than a Pathway problem. The pathway Text Edit wants to use has been corrupted and TextEdit has no idea where to turn.
Odds are VERY likely that if you look at the "start" folder of the path from which you are trying to save (and from which you probably save regularly), it is NOT the Top-Level Hard Drive Folder, but some folder inside the hard drive, possibly nested several layers in.
To Fix This:
1) Create any New TextEdit document, and click Save.
2) When the Save Dialogue box pops up, navigate BACKWARDS to the absolute Top Level Folder of your Mac (probably called something like "Macintosh HD" or the like - we are talking the level ABOVE your User name / account).
3) Click on that Top Level, and then navigate from there down through your User account to the actual folder where you would like to save your document. It does not matter how deep your desired folder is located as long as you Start from the absolute Top Level of your entire system.
4) Click Save. It should save Just Fine. 🙂
What you basically just did was tell your Mac which way to turn - the "directions" the Mac was using got corrupted and the TextEdit save process got lost in the woods. Doing the steps above sets the save process back on the correct path.
I have happily only ever had this problem twice, and this solved the issue 100% both times.