Does 「鳴く」 give any nuances about the sound being made? When we say "birds singing", there is a positive connotation. When we say "birds crowing", there is a negative connotation. "The birds are making some sound" seems to be objective, and thus neutral. I was wondering is it true that is absolutely neutral? In other words, would the sentence have a positive connotation, negative connotation, or is it an absolutely neutral sentence?

I think it is neutral; it just implies the subject is a bird, an insect, or some other kind of animal. For positive connotation, there is a verb `` 'chirp'. For bad connotation, there is a verb (not limited to bird but can be used more generally) `` 'make noise'.