A nose of a dog should read The nose of a dog or A dog nose. There should be some flexibility in these responses.
I would say "a dog's nose", as in "A dog's nose is an extremely sensitive organ." If I were using the word "human", both "a human nose" and "a human's nose" sound OK to me, but if I were using the word "man", "a man's nose" sounds OK, but "a man nose" sounds very strange. Native speaker of (American) English, but I can't explain it, only just what sounds right and what doesn't.