It certainly can be correct. As with all grammar it depends on how it is used
Yes, "she sings badly" is grammatically correct. The adverb "badly" is describing the verb "sings" in this sentence, indicating the quality of her singing.
It is a colloquial phrasing, because you are using "really" to mean "extremely" and "badly" to mean "desperately" or "greatly." A less colloquial form is very badly.
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yes it is a real word. It is also correct grammatically and more stupid is not correct.
"Real soon" is not grammatically correct. It should be "really soon" or "very soon."
Yes, 'for free', is grammatically correct.
"Not like that" can be grammatically correct, depending on the context.
Yes! That is grammatically correct!
This is not grammatically correct. The correct form is 'you do not know', or the abbreviated 'you don't know'.
"Missing" is not an action verb therefore should not be described by an adverb ( a word ending in ly). Likewise, "I feel badly" follows the same rule and is also grammatically incorrect.
'What a drunkard you are' is a grammatically correct English sentence.
"That was wrong" is a grammatically correct sentence.