Removing unused suggested names

If the single names are popping up, it means that you have some People tags with those unwanted names.  Here is what I suggest you do, using one of your examples:

• In the Tags hierarchy check the box for the People tag for Alex.  This will display all of the photos that have been tagged with that name.

• Select all of the photos in the grid (using Ctrl+A) and tag them with the Alex Houser People tag.  (If you don't have the desired tag name, create it or edit the Alex tag by right-clicking on the name without checking the box, and selecting Edit)

• Uncheck the Alex tag box and check the Alex Houser tag box to confirm that all of the tagged photos now have the correct tag.
• Right-click on the Alex tag and choose Delete from the context menu.  Dismiss the warning and click OK. 
• Wash, rinse and repeat for each unwanted name.

Now the photos originally tagged with Alex will only show a generic face icon in the named People room.   If you really want the faces to show in the People room, do the following:

• In the named People room, select the stack for Alex.
• Select all of the faces displayed for Alex (Select one of the faces and press Ctrl+A or Shift-click the first and last face)
• Click the Not this Person button (or choose Not Alex from the face's context menu)

This will remove all of the faces tagged with Alex from the Named People room and the photos should reappear in the UnNamed People room where you can tag them with the correct name.  You can then delete Alex from the People tag hiearchy in the Keyword tags panel.