I'd hyphenate over-the-moon type of feeling, but that's probably more because I hyphenate rather than quote. Quotes would work since it is a common phrase, but I'd add hyphens while I was writing and not go back and edit it.
Yes, it should be hyphenated. The normal phrasing should be "stamped with a Bates stamp," but that's too wordy. Bates is the brand name of the stamp. You probably already knew that. Anyway, it's a compound word created for the occasion, so to speak. Here's a site that talks about compounds. It seems to refer to the Chicago Manual of Style, which is my preferred reference book. http://www.docstyles.com/cmscrib.htm
Well, how do we do CAT? Computer-aided-transcription software or computer-aided transcription software? I think there could be an argument for both. The three words DO modify software. The first two DO modify design or transcription software.