Reply To: Where do I save *.INDT template files

InDesign templates can sit anywhere on your system there is no default location.
They will only stay in your ‘Open Recent’ if the template itself was one of the 20 most recent files (or what ever number you have set in preferences).

You can of course create a Template folder and save all your templates there.
When you want to use one just open the .INDT file in the finder or explorer and it will open in InDesign as a new unnamed document.
If you use the templates often make a Shortcut and place it in the sidebar (or equivalent on a PC) then you will see the shortcut in the Open dialogue box.