Possible, but unlikely. While the phrenic nerve descending in the neck carries branches that will enter the larynx to control the vocal folds, there are also axons that will branch to control heart/lungs in the cardiac plexus. Injury to the vagus would be expected to perturb cardiopulmonary reflexes, and as they are normal this would argue against the main branch of the vagus being injured.

Incorrect. Although the phrenic nerve is in the neck and of concern to protect during surgery, the nerve controls the respective side hemidiaphragm (e.g. left phrenic nerve controls the left hemidiaphragm). As the patients respiration is normal this argues against phrenic injury.   

Correct!  The recurrent laryngeal nerve branches from the vagus in the thorax and ascends to the larynx controlling vocal function. Since the nerve is in close proximity to the thyroid gland, it may have been injured or inflamed from the surgery and have diminished function causing the horse speech.

Incorrect. The sympathetic chain does extend into the neck, where it gives rise to the cardiopulmonary splanchnics and other nerves of the head/neck. Injury to the chain at this point would affect the cardiopulmonary splanchnics and thus cardiopulmonary Reponses. As these are normal in the patient sympathetic injury is not consist with the observations.

Incorrect. The cardiopulmonary splanchnics do arise in the neck and descent into the thorax, so they are in a location where injury was possible from the surgery. However, normal cardiopulmonary responses would suggest these nerves are functioning correctly.