Entities and Plugins. What's the relationship?
I am not sure that I quite get your question but in Drupal 8 things like nodes, files and users are entities. They all inherit from ContentEntityBase. Plugins are a way to implement/extend Drupal functionality such as Views filters and Field widgets.

From the docs:

Entity API


  Drupal 7 - entities were generic stdClass objects.
  
  Drupal 8 - entities are now specifically typed objects, with each entity type defining a class that will be used for instances of the given entity.
  
  Entity types in core comes in two variants.
  
  Configuration Entity
  
  Used by the Configuration System. Supports translations and can provide custom defaults for installations. 
  
  Content Entity
  
  Consist of configurable and base fields, can have revisions and support translations.


Plugin API


  Plugins are small pieces of functionality that are swappable. Plugins
  that perform similar functionality are of the same plugin type.
  
  Drupal contains many different plugins, of different types. For
  example, 'Field widget' is a plugin type, and each different field
  widget type is a plugin. The admin user may select from the list of
  field widget plugins to set the widget that a field uses.


After author edit:

Your specific use case would call for a event subscriber approach in my opninion. More specifically an event subscribed to the KernelEvents::REQUEST event. You can check at every page request if there are open Alert entities for the user and display them accordingly.

Your question as to if there are use cases for combined entities and plugins I wouldn't know. I only ever created Views and Block plugins and never combined entities and plugins. Perhaps you could create a Block plugin and show your Alert entities in there?