These are questions discussed by students of logic rather than by students of language as such.
Neither sentence is natural in everyday English. For #1 we would say, "Both are incorrect' or 'Neither is correct'. For #2, depending on the meaning we wished to convey, we'd say either 'Not all of the birds in the cage are blue / Some of the birds in the cage are not blue' or 'None of the birds in the cage is/are blue'.
Thank you for your reply. Sometimes I find myself spending much time dealing with sentences that don't sound natural in English. (Sentences that are offered as examples in grammar books.) I could immediately understand the meanings of the revised sentences you've offered.