After two days of searching for an answer to why template styles (classes) were not working in the JCE editor, this is the first one that worked. Also, the Styles dropdown box did not drop down and now everything is working fine. The solution came from Jenny, a Joomla guru or wizard or whatever. The point is, she got it right. Here's what she said:
I did it and it worked ... hope it does for you.You would have to set the editor.css file to be the custom css file you are loading with JCE. This is under JCE Administration -> Configuration located in the Formatting section.
You should set your Use Template CSS to "No" and Custom CSS file to "templates/$template/css/editor.css" (no quotes). When after you save that you should see your editor.css style selections.
One last thing: If there is no editor.css file in your template/css folder, you can easily create one with Notepad or similar text editor. Just list the classes that your template supports like this
.box {}
.check {}
.arrow {}
Just make a regular (vertical) list of them and save the file as "editor.css" then add it to your template/css folder. Then do what Jenny recommended above.