When I saved my Print Composer template, I cleverly navigated to a logical (to me) folder among my documents to save it. Now I find that it does not show up on the list of Composers under “File”. Can someone please tell me where it is supposed to be, so I can move it to there and open it?
It seems you are mixing up two different concepts:
Every QGIS project file can have multiple print compositions which are listed in Composer Manager. These compositions are stored within the project and don't have to be saved separately.
Additionally, you can create composition templates which can be reused in different projects. They are can be saved/loaded from within Print Composer and can be stored in any location you like.
To reuse a template, you first create (ADD) a new blank composer then import the saved one (FILE|LOAD FROM TEMPLATE). Since file locations are a local user issue, you will have capture that detail yourself. The LOAD FROM TEMPLATE has file navigation diaglog the same as the SAVE AS TEMPLATE.
Not the file as type, but it should default to *.qpt.