The question requires identifying the INCORRECT statement among the given options related to the spinal cord and sensory pathways.
The statement "Dorsal column fibres synapse at the cuneate and gracile nuclei" is true. These nuclei, located in the medulla, are the first central processing points for fine touch, vibration, and proprioception carried by the dorsal columns.
The statement "Pain and temperature sensations are carried by lateral spinothalamic tract" is true. This tract is the primary ascending pathway responsible for transmitting these sensory modalities from the body to the thalamus.
The statement "Spinal cord ends at lower level of $5^{th}$ lumbar vertebrae" is false. In adults, the spinal cord typically terminates at the conus medullaris, located around the level of the $L1$ or $L2$ vertebra. Below this, nerve roots form the cauda equina.
The statement "Dorsal spinocerebellar tract carries fibres from the same side of the body" is true. This tract transmits proprioceptive information ipsilaterally (from the same side) to the cerebellum.
Based on the analysis, the incorrect statement is that the spinal cord ends at the lower level of the $5^{th}$ lumbar vertebrae. The spinal cord terminates much higher, around $L1$-$L2$.
A 30-year old male was being evaluated for paraparesis. On examination, there was severe muscle atrophy, with absent deep tendon reflexes and absent Babinski sign. The most likely possibility is: