the adverb should be badly not bad i.e. I sing badly
Correct. DL probably put "bad" in dropdown menu to see if one knew the difference between an adverb and an adjective.
Dropdown contains all possible translations. Seems like both translations are valid - it just depends on the context
The words in the dropdown menus are not answers. They are hints.
"I sing badly in the morning" is correct here. In another place, the answer might be, "I felt bad yesterday."