Is it possible for a torn rotator cuff cause neck pain and headaches?

Yes: If you have a torn rc then commonly other muscles have to be recruited to help the shoulder to function more normally. This can include other muscles including those leading to your neck. This could theoretically contribute to neck pain and ha. However, you could also have separate neck pathology (arthritis, disc issues, etc) that are causing these symptoms. See your doctor.

Yes: There is some overlap with regard to cervical nerve roots in this pain distribution.