I have a qsave error (see first screen shot)
The simple answer is (and I think someone mentioned it...) that AutoCAD cannot delete a file that it needs to.
When you save a named drawing, AutoCAD:
1. Writes the content of the editor to a .TMP file.
2. Deletes the current .BAK file.
3. Renames the current .DWG file to .BAK
4. Renames the .TMP to .DWG.
If step 2 fails because you do not have delete rights, then the process stops and you end up with a TMP file (as your dialog indicates). You can replicate this behavior easily by saving a DWG in a directory in which you do not have delete rights.
As far as the DWL* files, these are created and managed by the WHOHAS.ARX file and have
*nothing* to do with AutoCAD saving (or not being able to save) a drawing.