Final answer:
The 65-year-old man with various neurological symptoms including crossed sensory loss, Horner's syndrome, and ataxia most likely has a stroke in the right lateral medulla, known as lateral medullary syndrome.
Step-by-step explanation:
The 65-year-old man presenting with acute onset of vertigo, nausea, vomiting, dysarthria, and nystagmus, as well as loss of pain and temperature sensation on opposite sides of his face and body, left Horner's syndrome, tendancy to fall to the left, and left finger-to-nose dysmetria is exhibiting classic signs of a lateral medullary (Wallenberg) syndrome. This clinical syndrome is most often the result of a stroke in the right lateral medulla of the brainstem. The symptoms given such as the crossed sensory loss, Horner's syndrome, and ataxic dysmetria point towards a lesion in the lateral part of the medulla oblongata on the right side, which affects the structures such as the spinal trigeminal nucleus, sympathetic fibers, and inferior cerebellar peduncle.