When an article includes a [[Category:Cat Name]], the Cat Name category is created, but the link shows in red. When one clicks on the category it goes to http://host.domain.org/index.php?title=Category:Cat_Name&action=edit withthe Editing Category window open.

The category tab is available, and if clicked on, the articles in the category do show up alphabetically.

Is there any way to make the default so that the contents of the category show up and not the edit screen?

Yes, this is the normal behavior. You must put content on the category page to prevent this--the content shows up at the top and the categorized articles underneath. It's used to give a summary of what sorts of articles one would find in the category. Try using   (a non-breaking space) if you don't want to write a description right away.

Also, categorizing the category page (with [[Category:Parent cat]]) will only display the article list but not give the edit page when users click the link.

Hínandil