1. "This is the chicken's comb." is correct. It is not incorrect to say "This is the comb of the chicken.", but it is the way a teacher might speak, and is more formal and academic.
2. "These are a bird's feathers." is correct. Again, it is not incorrect to say "These are the feathers of a bird", it is just more formal.
3. "These are buffalo horns." is correct. Again, "These are the horns of buffalo." is not incorrect, it is just more formal. By the way, the plural of buffalo is usually buffalo.
Also, your question was not grammatically correct. it should have been :
"If I point at a chicken, should I say, "This is the chicken's comb.", or "This is the comb of the chicken."? Or can I say both? The same mistakes in 2 and 3.