How nutty are the terms "nut case", "health nut" and "sports nut"?

If someone is nuts about something/someone it means they are a very enthusiastic— sometimes bordering on obsessive—devotee of that particular thing or person. To be nuts is a colloquial term meaning crazy, mad or insane, and there is the idiomatic expression to drive someone nuts, meaning a person's annoying or irritating behaviour is responsible for making another, figuratively speaking, lose their mind.

• When (and why) did the humble innocuous fruit, nut, become associated with insanity?

• We're all familiar with the politically incorrect term, basket case, I am referring to its modern-day meaning: someone or something that is incapable of functioning normally, a state of helplessness [...], most frequently in the context of mental health, is the idiom
  nut case therefore related to that expression or is it coincidental?

• Are the terms health nut and sports nut derogatory, neutral or positive today? Were they originally insults, as I suspect, or were they meant to be semi-affectionate, tongue-in-cheek expressions?

Nut is slang for head. And nut case means head case; i.e, mind/brain injury/illness.

It's a pretty obvious metaphor: nuts and human heads have breakable covers, are spherical living things, and are filled with other important living things (some of which may even resemble one another -- compare walnut meats with brain hemispheres visually, for instance).

Nut case, in turn, results in the adjective nuts, meaning 'crazy'. Then, by comparison, crazy about X becomes nuts about X; and someone who is nuts about X is of course an X nut.

Etymology shows that nut as a noun was first associated with insanity at the beginning of the 20th century, but as an adjective (nuts) it was used since 1846. Also the nut-case is first attested in 1959 that is much later than the origin of basket case. It appears there is no clear relation between the two expressions.

nut n.

Meaning "crazy person, crank" is attested from 1903 (British form nutter first attested 1958; nut-case is from 1959); see nuts.

nuts (adj.) "crazy," 1846, from earlier be nutts upon "be very fond of" (1785), which is possibly from nuts (plural noun) "any source of pleasure" (1610s), from nut (q.v.). Sense influenced probably by metaphoric application of nut to "head" (1846, as in to be off one's nut "be insane," 1860). Nuts as a derisive retort is attested from 1931.

I think that the expressions health nut has a derogatory meaning:

Health nut

A health nut is a person who is obsessed about his/her health. "Health nuts are going to feel stupid someday, lying in hospitals dying of nothing."

while sports nut is more neutral:

Sports nut:

A Sports fan. Someone who loves to watch sports. I love to watch sports. I go to soccer games all the time. I'm a sports nut.

I read that the probable derivation is that head became nut, because of the physical resemblance. Then "off his head" meaning crazy became "off his nut" which got shortened to nut.

Now that nut means crazy, most other usages "sports nut" derive from that.

EDIT: Confirmation that Etymology Online agrees with this on this post:

How did the phrase "are you nuts" come about?

The case of "Health nut" and "Sports nut"

I debated whether to edit my findings in my question or ask a separate question altogether. If I included the graphs in my question, it would seem I was less interested in the etymology and usage of nut=insane than I really am.

Anyway, after some reflection I am confident in stating that today health nut and sports nut are not derogatory terms, despite Urban Dictionary's sardonic entry. Although I can imagine someone labelling a person who is careful with their diet and does regular exercise as being a health nut, I feel that health freak is more pejorative and describes a health obsession taken to extreme more accurately. Generally speaking, the current health craze, which started in the 70s, has continued to gain momentum and anyone who cares about their physical health (and appearance) is usually admired.

Television began bringing teams [American football and basketball] from across the country into the spotlight. The exposure of sports continually grew, as did their popularity. Individual sports like tennis and jogging took off (Strasser & Becklund). The health craze swept the nation [...] . By the end of the 70s sports were becoming a marketing tool and industry on a national scale.

Of course I realize that the term sports nut may mean the person enjoys watching sports from the comfort of his/her couch, but the term doesn't preclude the possibility of them actually doing sports. In conclusion the term nut is used to denote someone's passion and enthusiasm rather than allude to their mental health.

To compare usage I added the colloquial terms: sports/health nut, sports/health freak, and sports/health crazy. The first chart shows a constant rise in popularity for health nut. AmEng Ngram

But if we use the same identical terms in the British English corpus, the results are quite surprising. Health freak surpasses health nut and the terms sports crazy and sports freak are non-existent! Why? I can only surmise that the general British public is more cynical and less sports/fitness/health oriented than their cousins across the pond.

BrEng Ngram

A Dictionary of Slang and Unconventional English by Eric Partridge (this 8th edition is dated 1982) confirms that "health/sport nut" has positive connotations.