why do we need to study english?

It seems "keep" has the connotation of physically holding something. Which would mean you can use "keep" best when it's associated with a physical object.

"Keep (the dog) happy/sad/healthy/fertile/beautiful" all work reasonably well because you have a physical object in mind.

The physical connotation also helps to give it a forceful tone. "Keep it secret. Keep it safe."

"Keep your health" is difficult because "your health" is much more abstract. Difficult for you to hold your own health, or beauty in your hands. "Stay" works better in that case because it's already a word that applies to a internal state rather than an external object.

"Keep the love." is bad but
"Keep the peace." Is good. (Invokes an image of protection, which is necessary to keep objects.)

I think idioms that use "keep" also exploit the idea of turning abstract concepts into a physical ones ("Keep your wits about you.")

You can also have "keep" mean "persist":
Keep your eyes on the prize.
Keep truckin'.
Keep trying.


(As a side note, I don't think "Keep (subject optional) ______" is not really a thing. Leaving out the subject just means that the imparative "you" takes the role of the subject. "Keep quiet" is "You keep quiet")