Is the Custom CSS Manager plugin the same thing as creating a child theme?
Almost, but a Child Theme allows much more customisations like changes to PHP and HTML.
If you have a Child Theme, you do not need to download the Custom CSS plugin. You can use the style.css file of your Child Theme.
Also, If I were to create changes to other files such as functions.php, then I would have to do it by creating the child theme folder I would assume.
Yes, that's right.
But I can leave the style.css out of the folder because it's being managed by Custom CSS Manager?
No, sorry, you'll still need your style.css file in your Child Theme.
You should either use a Child Theme or Custom CSS plugin.