The use of south as in the phrase go south stems from the 1920s (from the Oxford English Dictionary):
colloq. (orig. Stock Market). Downward or lower in value, price, or quality; in or into a worse condition or position. Esp. in to head (also go) south.
1920 Elgin (Illinois) Dairy Rep. 13 Nov., Meat, grains and provisions generally, are like Douglas Fairbanks, headed south—in other words, going down.
The reason South is correlated with down is because of its use in the standard Western set of cardinal directions:
According to the Oxford English Dictionary, the meaning of telling someone to "go west" is not related to the phrase "going south". Instead, it means:
go west, young man: used as an encouragement to seek fortune in the American West; also in extended use.
Attributed to Horace Greeley, who, according to Josiah Bushnell Grinnell, gave the latter this advice in September 1851 (see quot. 1891)
The phrase go west on its own also refers figuratively to death (as the sun sets in the West).
There is no similar idiom "go east", unless one is actually telling someone to go in the cardinal direction of east.
The direction "north" has a figurative meaning as well:
Higher; esp. in north of (a figure, cost, etc.): higher than, in excess of.
Though north and south have related meanings (higher and lower, respectively), they are used in different ways. North is usually used in respect to a give figure or amount. For example,
2001 San Francisco Business Times (Nexis) 9 Nov. 21 What's your average deal size? It's gone north of $250,000 per contract even as high as $300,000 per contract.
However, south is usually used as a general figure of speech. For example,
2003 R. B. Parker Stone Cold (2004) xl. 154 But your marriage went south and you had a drinking problem.
That being said, the two can be used both to refer to figure amounts, although in my experience this usage is rarer:
1986 Financial Times (Nexis) 5 July i. 6 With oil heading south of $10‥the London stock market today stands less than 4 per cent below its highest ever level.
According to Eric Partridge's A dictionary of slang and unconventional English, Hong Kong dog is a tropical fever, and the term is originally 20th century Royal Navy slang.
His source for this is 'Taffrail', or Capt. H Taprell Dorling, DSO, RN, in Carry On, 1916, especially the article 'the Language of the Navy, originally published not later than 1915.
The term is comparable with Malta dog, another local name for traveller's diarrhoea, is also from Royal Navy sailors.
Hong Kong dog can be found as far back as 1899. The Philadelphia medical journal says:
Sprue (psilosis linguae, Hong-Kong dog, Ceylon sore mouth, etc.) is a chronic catarrh of the alimentary canal from the mouth to the anus, accompanied by tenderness of the tongue, diarrhea of a special character, and an atony and ...
Also from 1899, Madam Izàn: a tourist story says:
Cholera at Port Said, leprosy at Colombo — I heard of a family who had caught it in their washed linen — the plague here. Then the fog. And besides that, a fever they call the Hong Kong dog, which is nearly as bad as the plague
William Ernest Russell Martin's 1924 The adventures of a naval paymaster has a whole chapter on Hong Kong dog, unfortunately not readable via Google Books.
But why dog? According to slang lexicographer and author of Green's Dictionary of Slang:
The primary divisions are place-related, images of dancing or fast movement [e.g. Aztec two-step, Greek gallop, Rome runs, Tokyo trots], and rhyming slang usually based on 'the shits'.
As for dog, there's anecdotal sources describing either being "bitten" by the dog, or the onomatopoeic "barking" noises you made in corners when you were suffering from it.
At All Costs by Sam Moses describes Malta dog during WWII:
Sand flies flew out of the cracks in the limestone, carrying their fever. Cockroaches, bedbugs, and lice ruled. The "Malta Dog," a virulent form of dysentery, barked in the corners. The hunger never went away.
Band of Eagles by Frank Barnard also describes WWII pilots falling sick:
"Bitten at last by the Malta Dog"
Some forum postings agree:
The "Malta Dog," a virulent form of dysentery, When you got it you barked in the corners.
Malta dog - was what you were bitten by in flight the morning after a night in downtown Valetta in "the Gut", after consumption of copious quantities of Cisk or Hopleaf beer; or (God forbid), Farmers' Wife wine. Usually comprised of the runs interspersed with the odd projectile vomit, both accompanied by the "cheese-wire round the forehead" headache. If you had a good aim you could throw up out the beam lookout window of a Shackleton with little risk of blow-back. Thank the Lord those days are over.
Malta was the headquarters of the Royal Navy's Mediterranean Fleet from 1814 until the mid-1930s, so there was a large navy presence. This ties in with the Royal Navy roots in Partridge's etymology for Hong Kong dog.
Best Answer
The original question asked about the idiom put on a clinic to mean "to perform exceptionally well." While the term originated with medical clinics and was transferred, as OED notes, to non-medical sessions "for instruction in or the study of a particular subject", the idiomatic use of demonstrating how to do something well through performance, rather than through formal instruction, came later.
It has been used extensively in sports. The earliest I've found is a 1936 article about a pitcher with the Little Rock Travelers of the minor league Southern Association in a game against the Birmingham Barons: