**Core Concept**
The patient's symptoms suggest a neurological disorder that involves the brainstem and cranial nerves. The presence of impaired sensation on one side of the face, ataxia, nystagmus, dysphagia, and hoarseness of voice, along with impaired pain and thermal sensations on the opposite side of the face and Horner's syndrome, indicates a lesion in the brainstem that affects multiple cranial nerve nuclei.

**Why the Correct Answer is Right**
The symptoms described are characteristic of a lateral medullary syndrome (also known as Wallenberg's syndrome), which is caused by a thrombosis or embolism in the vertebral artery or the posterior inferior cerebellar artery (PICA) that supplies the lateral medulla. This leads to damage to the nuclei of the vagus nerve (cranial nerve X), the spinal trigeminal nucleus and tract, and the sympathetic fibers that give rise to Horner's syndrome. The impaired sensation on one side of the face is due to damage to the spinal trigeminal nucleus and tract, while the ataxia and nystagmus are due to damage to the cerebellar connections.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because the anterior inferior cerebellar artery (AICA) primarily supplies the anterior inferior part of the cerebellum and does not typically cause a lateral medullary syndrome.

**Option B:** This option is incorrect because the superior cerebellar artery (SCA) primarily supplies the superior aspect of the cerebellum and does not typically cause a lateral medullary syndrome.

**Option D:** This option is incorrect because the basilar artery supplies the brainstem, but a thrombosis in the basilar artery would more likely cause a pontine or midbrain syndrome rather than a lateral medullary syndrome.

**Clinical Pearl / High-Yield Fact**
The lateral medullary syndrome is often caused by a thrombosis or embolism in the vertebral artery or PICA, which can be precipitated by conditions such as atherosclerosis, hypertension, or vasculitis. A high index of suspicion is required for early diagnosis and management, as the symptoms can be subtle and may progress rapidly.

**Correct Answer:** C.