Edit: found the citation from 1672, from Andrew Marvell’s The Rehearsal Transpros'd:
Two or three brawny Fellows in a
Corner, with meer Ink and
Elbow-grease, do more Harm than an
Hundred systematical Divines with
their sweaty Preaching.
It's also defined in B.E.'s A New Dictionary of the Terms Ancient and Modern of the Canting Crew, in its several tribes, of gypsies, beggers, thieves, cheats, &c. with an addition of some proverbs, phrases, figurative speeches, &c., c.1698:
Elbow-greaſe, a deriſory term for
Sweat. It will coſt nothing but a
little Elbow-grease ; in a jeer to one
that is lazy, and thinks much of his
Labour.
I found no earlier mentions than senderle, but here are some useful references. These are the earliest references I could find, and helpfully, they are also dictionary definitions.
The Online Etymology Dictionary says
Phrase elbow grease "hard rubbing" is
attested from 1670s, from jocular
sense of "the best substance for
polishing furniture."
There's a similarly colourful definition in Francis Grose's 1785 A Classical Dictionary of the Vulgar Tongue:
ELBOW GREASE, labour, elbow grease
will make an oak table shine.
(The rest of this dictionary is interesting too!)
Also, very pertinent to the question, here's The Royal Dictionary, French and English, and English and French by Abel Boyer in 1729:
Elbow-grease, (or Pains) Rude travail.
Rude travail is French for rough work. There's no entry for "l'huile de coude" in the French side.
And in John S. Farmer and W.E. Henley's 1905 A Dictionary of Slang and Colloquial English:
Elbow-grease. Energetic and continuous
manual labour : e.g. Elbow-grease is
the best furniture oil : Fr., huile de
bras or de poignet ; du foulage
(1779).
French huile de bras or de poignet is oil of the arm, wrist which is quite close. I think du foulage is fulling, the manual scouring and milling of cloth.
The earliest French reference I could "l'huile de coude" helpfully explains the term. In Jean Humbert's 1852 Nouveau Glossaire Genevois: Volume 1 (New Geneva Glossary):
Dans le langage badin des domestiques
et des maîtresses, l'huile de coude,
c'est le frottage, c'est-à-dire : Le
travail de la servante qui frotte.
Ces meubles, Madame, ne veulent pas
devenir brillants. — C'est que, ma
mie, tu y as sans doute économisé
l'huile de coude; c'est-à-dire : Tu as
trop ménagé ton bras et tes forces.
A rough translation:
In the playful language of servants
and masters, elbow grease is rubbing,
i.e. the work of the maid
who scrubs. This furniture, Madam,
does not want to shine. - My dear,
that is because you have undoubtedly
skimped on the elbow grease. In other
words, you have conserved both your arm and
your strength.
These references also suggest that "l'huile de coude" is an anglicisme.
Best Answer
The Ngram chart in the OP's question suggests that "dog eat dog world" first appeared in print in 1954, and that "doggy dog world" first appeared in 1984. A Google Books search, however, finds earlier instances of both phrases.
1. In the jungle out there, do dogs normally eat dogs?
In fact, the phrase appears to have arisen first in the context of an observed disinclination of dogs to eat dogs, as Christine Ammer, American Heritage Dictionary of Idioms (1994) points out:
But the sense of the Fuller quotation and the Roman saying are much to the same effect: that dog eating dog is highly unnatural and therefore a sign of an environment in extremis. The notion that, in human relations, "it's a dog-eat-dog world out there"—which presents (figurative) canine cannibalism as an essential aspect of the normal state of the world—is thus fundamentally at odds with the original sense of Fuller's saying.
In any case, the phrase "dog-eat-dog world" shows up in Google Books search results as early as 1918. From Elias Tobenkin, The House of Conrad (1918):
2. When and why did 'doggy dog world' emerge?
As for "doggy dog world," the first Google Books match is from U.S. Senate Committee on Veteran Affairs, Educational Benefits Available for Returning Vietnam Era Veterans (1972) [snippet]:
The rise of "doggy-dog world" in the mid-1980s seems to have occurred independently of the instance recorded in 1972. Andrew Tobias, Money Angles (1984) [combined snippets] introduces the wording in the context of a child's misunderstanding of the phrase "dog-eat-dog world":
To similar effect, from an item on phrase misunderstandings in New York magazine, volume 17 (1984) [combined snippets], we have this:
And from The Esquire Fiction Reader (1985) [combined snippets]:
"It's a doggy-dog world" is thus simply a rewording of "It's a dog-eat-dog world" in misheard and misparsed form. The 1984 upsurge in published occurrences of the phrase probably reflects not a sudden spike in real-world confusion over the wording of the phrase, but either coincidental observations of the mistake or the spread of such observations from one published source to another. The matches from Google Books do not include any appearance of "doggy dog world" in a bestseller or other source that is likely to have triggered mass awareness and usage of the phrase.
It also bears observing that in 1984, the year when, according to Ngram, a measurable frequency of use of "doggy dog world" first arose, the phrase "dog eat dog world" was roughly ten times more common, whereas in 2000 (the last year tracked in the Ngram chart), "dog eat dog world" is now almost twenty times more common. So in relative terms, the success of "doggy dog world" over the first sixteen years following its abrupt arrival on the scene was less than overwhelming.