Nerve in the singular has several metaphoric meanings, including "courage, boldness" at definition 4. It is often, though not always, used in the phrase "to have a lot of nerve," which is more definition 7 and is not a good thing to say about or to someone: "You have a lot of nerve walking back in here after what you said last night!"
But nerves in the plural means "agitation, anxiety" at definition 9. To my (American) ear "have nerves" sounds rather dainty and euphemistic, something a character in a Victorian novel would say rather than a common turn of speech in modern English. Others may have a different opinion.
Run [adjective] means to tend to be [adjective], as you said. Usually I would associate this usage with a description of a piece of machinery or other non-living being, like "this engine runs hot" (meaning: "this engine is hotter than you would expect based on the normal state of similar engines"). So to say someone runs highly anxious means they are more anxious than the baseline. This is in line with "having nerves," not an exception to it.