With regard to early occurrences of (versions of) the expression, I note this poetic (but literal) example from "Texas and Oklahoma," in the [Sacred Heart, Oklahoma] Indian Advocate (November 1, 1909):
Then the Lord sent the zephyrs to blow day by day / And temper the heat of the sun's golden ray, / And he hung the moon low that its silvery light / Might fall like a dream on the sweet-scented night.
But use of the expression to indicate an especially admirable and capable human being appears less than a decade later. From Dorothy Dix, "Girls Who Are Dangerous," in the El Paso [Texas] Herald (July 22, 1916):
Then there's the girl who's just girl, and looks as harmless as a kitten. She wears white, fluffy things, and she looks up at you as if you had hung the moon and made a good job of it. and she's so little, and clinging, and helpless, that a man begins to pet her as instinctively as he would a child.
Another early instance appears in Dorothy Dix ("The World's Highest Paid Woman Writer"), "Why Men Prefer Second Rate Women?" in the Honolulu [Hawaii] Star-Bulletin (June 25, 1917):
Generally speaking the happiest marriages are those in which the wife is not in the husband's class intellectually, and where she prefaces every sentence with "John says" and thinks he hung the moon instead of having taken a course in astronomy.
It begins to look as though Dorothy Dix may be single-handedly responsible for this idiomatic usage. In ny case, it appears in other writers' work long before 1953. From Oren Arnold, "Saddle Strings," installment 13, in the San Bernardino [California] Daily Sun (May 26, 1939):
"We've got acres of room, xtra beds. Or you could sleep with Don like you used to. Don still thinks you hung the moon."
"I did. See It out there?"
Sure enough, the moon had been hung. The taxi driver, deducing what was expected of him, had ambled up the beautiful drive that skirts Camelback mountain, and now the moon was a theatrical spotlight over the hump of the camel itself.
From "We, The Women," in the Borger [Texas] Daily Herald (June 24, 1940):
Every college ought to offer its women student a course called "How to make your husband shine."
Taking it woul be the best marriage insurance a girl could have.
She would be told such things as:
In speaking of him or of his work, take the attitude that you are talking of a man of importance, whose ideas and opinions are of consequence. You call scarcely over-do such an attitude, for it is much better to have people say. "She thinks her husband hung the moon" than "She's a swell person, but I don't think he amounts to very much."
From "Stuff," in the [Houston, Texas] Rice [University] Thresher (December 6, 1940):
Seen or scene on the campus: What about this guy in Denver that thinks Adams hung the moon? . . .
And from Clark McMeekin, Show Me a Land, serialized in the [Honolulu, Hawaii] Nippu Jiji (November 24, 1941):
Miss Spicy was airing blankets in the back yard. "Your young man has gone in town with the doctor," she told Dana with triumph in her eye. "He's been good to Dr. Colly, I must say. I got it straight from the Pattersons that he paid off the mortgage on this place. . . . The doctor thinks he hung the moon. Wouldn't it be funny if he stayed here? He’d be like a son to Dr. Colly."
Now, to address the posted questions directly:
1. How old is this American English idiom?
The phrase appears as an idiom in Dorothy Dix's writing in 1916 and again in 1917. Dix (Elizabeth Gilmer) was born in 1861 on a plantation on the border between Tennessee and Kentucky, and she spent the years between 1896 and 1923 as a newspaper columnist for the New Orleans [Louisiana] Daily Picayune. So if she didn't invent the expression "hang the moon" herself, she probably heard it being used in the U.S. South, where she grew up and lived as a young adult, and it seems very likely that she helped to popularize it.
The matches for the term during the period 1939–1941 seem heavily weighted toward Texas, California, and Hawaii, but (as I've noted on other occasions) Elephind search results from the period after about 1926 skew toward those states because the Library of Congress's newspaper database for the United States as a whole ceases to be searchable for newspapers from more recent years. That leaves the Texas and California state newspaper collections (in particular) to be disproportionately represented in Elephind search results from the 1930s and later.
I have lived most of my life in Texas and California, and I don't recall ever having heard anyone use the idiom "hangs the moon" in the way that the posted question asks about. Having said that, I don't think McCaskill is using the phrase sarcastically; rather, I think she really means "I hope [Ocasio-Cortez] has a brilliant career"—although I imagine she also hopes that Ocasio-Cortez will moderate some of her more leftist views and become a more mainstream Democrat. In that respect, "I hope she hangs the moon" isn't obviously double-edged in the way that, for example, "bless her heart" commonly is.
2. Is this idiom reserved for someone you really admire or is it used to express incredulity and skepticism, think of the adage "and pigs might fly"?
It can appear in either context, I suppose. There is an inherent note excessive admiration in ascribing the ability to hang the moon to any mortal, and such excess is easily turned to a sardonic purpose.
Consider this allusion to a song by country music legend George Jones in Joe Billy Greasemonkey, "In Fond Praise of Redneck America," the Abilene [Texas] Optimist (December 2, 1983):
[John] Anderson isn't the only country singer singing about love. George Jones king of the she-broke-my-heart-so-I-turned-to-drinkin' country ballad, contributes his philosophy of love by singing "I think she hung the moon...upside down." Nobody, especially the "Stray Cats," can top the "Possum's" heartrending "I Always Get Lucky With You."
The reference here is to Jones's song "She Hung the Moon" (1983), which expresses considerable ambivalence about the wonderful qualities of his beloved:
You think that I think she's perfect / And she can do no wrong / Well, that's partly true, I don't think I do / But I've loved her for so long
I think she hung the moon upside down / And I think she's perfect at turning my world around / And if your heart, it needs breaking / She's the best I've ever found / She hung the moon but she hung it upside down
They say I put her up there / On a pedestal so tall / Well, I'll hang around down here on the ground / Just to catch her in case she falls
I think she hung the moon upside down / I think she's perfect at turning my world around / And if your heart, it needs breaking / She's the best I've ever found / She hung the moon but she hung it upside down
Notwithstanding occasional ironic use of the phrase by Jones and others similarly situated, it seems clear that normally describing some particular person as having hung the moon expresses deep admiration for (if not infatuation with) that person.
3. What would be a British English equivalent?
I can't think of a specifically British counterpart to "hung the moon." More broadly in English, people say and write things like "X lit up [or 'brought light to'] my world"—a comparable exaggeration, but one that seems so commonplace (bordering on cliché) that we rarely think about its literal meaning.