The surgery on the neck should have nothing to do with a dry throat and a cough, as the surgery did not even involve those structures.
It could have something to do with pain in the shoulder, however, because if there is an issue at C4-5, or the level above where you had your surgery, that can cause shoulder pain. This would be evaluated by a MRI of the cervical spine.
And the fact that the index finger has not returned to normal sensation probably means that there was some permanent damage that occurred to the nerve prior to the surgery (due to the nerve being compressed). That would likely be the C6 nerve, and if it hasn't improved this far out, that is likely the way it is going to be.