Are .module files still required? I noticed that, in the current page crumb module, there is no .module file. 

Why is that no longer necessary? As a test I added a .module file, but it seems it is not being read.  Where can I find documentation about using or not using a .module file in Drupal 8?
.module files are not required if you are not adding any procedural code. If you need to use any hooks, like hook_theme or hook_form_alter, yes, you need to do that in the .module file, because it cannot be done anywhere else. A lot of modules that extend base classes or provide services don't have any need to also include a .module file unless they provide something via hook_help, for example.

Some developers provide empty .module files, like:

&lt;?php

// empty


I simply omit them if they have no code.


  And as a test I added a .module file but it seems it is not being read.


You'd have to elaborate on this, because any module that is enabled that has a .module file is definitely read. You may have to clear cache to pick up on changes.