The monkey, who I wonder if it can see my ears.

Disclaimer: I have Cs in English Class… 😉

If your wondering isn’t critical to the thought I would simplify to “the monkey who might be able to see my ears” or “can possibly see my ears”. Otherwise AbouBenAdhem has the best option, though I might also suggest “The monkey whose ability to see my ears I’m wondering about”; splitting the prepositional phrase is more strictly proper but I think it reads a little better like this.

None of them are grammatically correct because none of them are complete thoughts let alone sentences.

All three try to specify the particular monkey by enumerating that it can see your ears but do no more.

Take away the description of the monkeys ability to see your ears and what you’re left with is “the monkey”.

“The monkey” isn’t a sentence.

If you are the subject and what’s happening is that you’re wondering if the monkey can see your ears then the sentence you want is “I’m wondering if the monkey can see my ears.”

If, as I suspect, you’re using “the monkey whose ability to see my ears I’m wondering about” as the subject of some larger more complex and cool sentence then you gotta lay out that part before someone can give solid grammatical advice.

They do all sound stupid.

As it’s a complete statement just say “I’m wondering if the monkey can see my ears.”

Because it is ‘the’ monkey, rather than ‘a’ monkey, it is implied that the monkey has already been referred to.