I think that RaceYouAnytime and lbf have satisfactorily settled the first two questions posted above.
On the the third question—involving when derangement in the sense of insanity first emerged in English, I did a bit of research into early instances of it and of deranged in a similar sense. I found this relevant instance from "Anecdotes of Pope Ganganelli," in The Bee or Literary Intelligencer (August 10, 1791):
It is very well known in Rome, that towards the beginning of this century, in the pontificate of Clement XI. of the Albani family, some Scots fanatic religionist took it in his head to covert the Pope ; accordingly, he walked to Rome, and one day that the Pope was going to the altar, carried high in his chair, in his pontifical dress, the zealous minister bawled out as loud as he could, Papa, habeo aliquid tibi dicere ex parte Dei. "Pope, I have a message to you from the part of God." Many thousands were present, as it is usual, in St. Peter's church, in great solemnities. Cardinal Hannibal Albani, the Pope's nephew, with great presence of mind, took one of the Swiss guards with him, to open the way directly through the crowd towards the place from whence the voice came, met the mad-man, and aid to him, Ostende mandatum ; "Shew me your credentials." The Scotsman had no more to say, was put into custody, was treated charitably, as a person deranged in mind ; his rags were changed for a good new coat, and was let go about his business.
Slightly earlier is this instance from a review of Thomas Howel's A Journal of the Passage from India, in The Critical Review, Or, Annals of Literature (December 1789):
Our author gives a short account of his journey from Madras across the peninsula of India ; but from this we can extract nothing very interesting. One of his companions, Meer Joad, whose passionate conduct seemed to betray a deranged state of mind, is frequently introduced in the journey from Bassora to Bagdat:
The noun form derangement appears in a British chancery court decision, The Attorney General against Parnther and Others (decision rendered sometime between February 1792 and July 1792):
The cause came on to be tried before Lord Kenyon, and a full special jury, on the 14th December , when a great deal of evidence was given in support of the plaintiff's case, by the persons who attended on Mrs. Barker, to prove general derangement, though with intervals of sense. ...
If derangement be alleged, it is clearly incumbent on the party alleging it, to prove such derangement : if such derangement be proved, or be admitted to have existed at any particular period, but a lucid interval be alleged to have prevailed at the period particularly referred to, then the burden of proof attaches on the party alleging such lucid interval, who must shew sanity, and competence at the period when the act was done, and to which the lucid interval refers ; and it certainly is of equal importance, that the evidence in support of the allegation of a lucid interval, after derangement at any period had been established, should be as strong and as demonstrative of such fact, as where the object of the proof is to establish derangement.
It thus appears that deranged in the sense of "disturbed [state of mind]" was in use by December 1789, and in the sense of "insane" by August 1791; and that derangement in the sense of "insanity" was in use—and had some degree of legal status as a term describing mental incapacity—in the British High Court of Chancery by July 1792. That date for derangement is eight years earlier than the first date cited by the OED, as reported in lbf's answer.
Update (October 20, 2018)
RaceYouAnytime's discovery of an early British instance of "mental derangement" from October 31, 1788, prompted me to make a foray into the British Newspaper Archive (a service to which I am not a subscriber). In reviewing the thumbnail snippets there, I found an instance of "derangement of ideas" from 1780 that seems to point to mental disorder rather than merely to disappointment of a person's plans. From "London, Nov. 6," in the Stamford [Lincolnshire] Mercury (November 9, 1780) [combined snippets; correction of OCR errors based to some extent on guesswork]:
Cox, the Attorney, now under of [?] in Newgate, is said to have lost his senses; at least his conversation is of such a nature, as to indicate a total imbecility of mind, derangement of ideas.
This same newspaper database returns a version of the story cited in RaceYouAnytime's answer from the [London] Times of October 31, 1788, reprinted in the [Midlothian, Scotland] Caledonian Mercury of November 3, 1788. It also returns a flurry of instances from the same year involving the madness of King George III. The earliest of these instances refers delicately to "the present unhappy derangement of his Majesty's health," in The Scots Magazine (November 1788):
His Lordship having delivered these few Sentences, sat down, and the House remained in a deep silence for a short space—when Earl Camden rose, and, in a low tone of voice, observed, that the present unhappy derangement of his Majesty's health, and the situation into which the public affairs were thrown, was a new event the history of this country.
But subsequent instances use the word derangement independently of the word health. Here are several instances of this usage from the period between November 13 and December 1, 1788.
From "London," in the Derby [Derbyshire] Mercury (November 13, 1788):
The Physicians are afraid, from the above Circumstance of his Appetite in the Delirium, and from other unfavourable Appearances, that the Fever is the Effect of the Distemper in the Head and not the Cause. In his Memory, too, he seems to be tenacious of Circumstances and Names, but the much-apprehended Derangement is apparent in the Application. Even in those Moments when he is most ungovernable, he retains his common Recollection ; that is, he will eat and drink in a rational Manner, and make pertinent Remarks on the ordinary Circumstances of the Table, or the Chamber. In every other Respect, he is most lamentably Deficient.
From "The King," in the [Kent] Kentish Gazette (November 14, 1788):
The result of their deliberations were Tent in the afternoon to the Prince of Wales at Windsor. His Royal Highness, it is thought, will be declared Regent, should the King not recover his faculties, which, we are sorry to add, the physicians are much afraid of. Cases of derangement remaining in the Monarch, the Constitution allows the Heir Apparent to act as Regent.
From "Saturday's Post," in the Hampshire Chronicle (November 24, 1788):
Yesterday the King was so collected, that he sent for Mr. Mfkiendie [?], who read to him for hours. He expressed his opinion that, his derangement, for so he himself termed his disorder, was not violent.
From "Thursday's Post," in the Hampshire Chronicle (December 1, 1788):
His time passes in conversation with the attendants about his person; and during the whole of yesterday, he displayed a strength of mind not equalled any time during his present unhappy derangement. About eleven o'clock on Monday, the Princesses were walking the garden. His Majesty observed, and immediately knew them.
It thus seems quite possible that use of derangement to refer to mental illness first became widespread when journalists and attendants of King George III adopted it as a euphemism for madness in 1788. At that time of course, there was no thought of using "George III derangement syndrome" in an ironically dismissive way.