What is the historical/etymological connection between nerves and the litany of expressions pertaining to anxiety, agitation, and audacity?

The sayings you are referring to appear to derive from the neurological sense of nerves:

From the neurological sense come Nerves "condition of hysterical nervousness," attested by 1890, perhaps from 1792. to get on (someone's) nerves is from 1895. War of nerves "psychological warfare" is from 1915.

The “audacity sense is attested from the 17th century:

The secondary senses developed from meaning "strength, vigor; force, energy" (c. 1600), from the "sinew" sense. Hence the non-scientific sense with reference to feeling or courage, first attested c. 1600 (as in nerves of steel, 1869) and that of "coolness in the face of danger, fortitude under trying or critical circumstances" is by 1809.

As far as I know similar usages are common in other European languages such as French and Italian.