I find that I have ribbons in all ribbon levels accumulated. Where do I find how they are earned and posted in the achievement box? (The icons, such as "Stable Awards", seen on the Stable Page.

If you're talking about the little blue, red, yellow and white ribbons that say 1st - 4th, those aren't awards. Those are earned from showing your horses, and you get a ribbon added every time your horse places.

So how are the icons listed on the Stable Page?

If you're talking about the featured awards, they get awarded at the end of each game year. Once you've earned an award you can click the "Award" button beneath the ribbons. This will show you all awards you've earned sorted according to the game year you've earned it. There you can chose to feature or unfeature one.