A C7 cervical spinal lesion means that the spinal cord or nerve roots are involved at that level. The damage to the roots cause inability to use the triceps as you stated and also some problems with sensations as detailed below. However, a deeper lesion may damage posterior tract fibers and thus the effects are seen below the level like C8 and lower levels.
Some details are as here with:
Levels of lesions: The level C7 mainly forms the radial nerve to innervate triceps muscle. This C6-C7 segment is susceptible to degeneration, trauma, and intervertebral disc dysfunction/ herniation, and fractures at this level.
Sensory: The C7 dermatome includes areas of skin covering the shoulders, back of the arms, and the middle finger. Findings include numbness, tingling and loss of sensation.
Motor: Motor functions supplied by this segment include straightening the elbow, lifting the wrist, elongating the fingers to an outstretched hand, and the triceps muscle in the upper arm. Weakness of these muscles are found.
Clinical Tests: 1. A reduced or absent triceps reflex may be indicative of a lower motor neuron lesion affecting the C7 nerve root.
2. Test sensation in the specific dermatome.