Instead of “half of an apple is eaten” or “an apple is half eaten,” can I say, “half of an apple is half eaten”?
No. The three sentences all mean different things.
Half of an apple is eaten means there was one half of an apple, and all of that one-half is eaten. Picture someone cutting an apple into two halves; one half has been put aside and is not being discussed. The other half is what we are talking about, and it is entirely eaten.
An apple is half eaten means there was one whole apple, and half of that whole is eaten. Picture someone starting with a whole apple and biting into it—they have only eaten half so far.
Half of an apple is half eaten goes back to the first situation: There was one-half of an apple, and half of that half is eaten, meaning one-quarter of the original whole apple.
Additionally, none of the sentences really sound right to my ear. If you are describing a scene There is a half-eaten apple sounds better; if you are describing the apple specifically, Half of the apple has been eaten.