Is there any convention about the form IDs? Is there a convention for the form IDs? 

I see that core sometimes uses module_name.form_name and sometimes module_name_form_name.
In my case, I'm using module_name.form_name.
Drupal core doesn't use the dot in the form IDs because, as the documentation for FormInterface::getFormId() says, the form ID should be a unique string that can be a valid PHP function name, since it's used in hook implementation names such as hook_form_FORM_ID_alter().

The form IDs Drupal core uses follow the &lt;module name&gt;_&lt;form name&gt; schema, so you will find form IDs like the following.


user_admin_settings
action_admin_manage
layout_builder_add_block


As long as the form ID is unique, any scheme would work. The convention to prefix an ID with the module name (i.e. the module machine name) has been used from Drupal since many years.