A: If you had a pet cockatoo, what would you call it and why?

 I think it will be different by their behaviors or colors. 

*I think it would be different based on their behaviors and colors. 

~> Corrected: The sentence is corrected grammatically.​

B: Make these sentences into English.

-난 뉴욕으로 가는 비행기를 예약할 거야. I will book a plane to New York.

*I will book a flight to New York.  

~> This sentence is okay. This is another way of saying it.

-이 연고는 간지러움을 덜어준다.  This medicine relieves the tickling.

*This ointment relieves itching.

~> Your sentence has a correct grammatical structure, but this has the correct usage/expression.

-그들은 새로운 도서관을 세웠다.  They build the new library.

​*They erected / built a new library. 

~> Corrected: The sentence is corrected grammatically.​