"Do a song" (노래를 하다) is literally how they say sing (노래하다, effectively a less formal version of the other if anything) in Korean. But even the dictionary gives "sing a song" for this, so I haven't a clue why they wouldn't accept it . . .
먹어보세요 means "try to eat" and 해보세요 means "try to do." They are both suggesting that a person should try something.
---This bulgogi is really delicious! Try it. 이 불고기는 아주 맛있어요! 먹어보세요!
---Riding a bicycle is fun. Try it. 자전거를 타는 것이 재미있어요. 해보세요.
They are both fine grammatically in English, though the meaning is slightly different. Literally this is "Sing and see" so I think it's closer to "Try and sing." "Try to sing" suggests something more like 노래를 부르려고 to me.