WordPerfect used to let me save poems with extension ".pom" and letters as ".ltr". Word will let me insert those extensions (e.g., "landlord.ltr.docx" and "TabbyCat.pom.docx" But is there a way I could have my Word letterhead template add the ".ltr" and my poem template add ".pom"?

You could but:

1. You could not use Save As in word to get that, you would have to modify the extension in the operating system.
2. You would have to tell Windows to open such files in Word.
3. These may conflict with other programs.

Word is not WordPerfect. See Word for Wordperfect Users.

The reason for this dates back pre-Windows to DOS where filenames were restricted to the 8.3 format. With Windows, you have the ability to use much longer filenames. See Naming Files - A System is the Key. The original of that was written for Wordperfect users in the 1980s.

Your letterhead should be a template.
Setting Up Letter Templates - Ribbon by Suzanne Barnhill, MVP