Learn English – “too dumb not to ~”

Indeed, ever and anon goes back at least as far as Shakespeare, who used it in Henry IV, Part 1. Wikipedia says it "was almost certainly in performance by 1597" and "was entered into the Register of the Stationers Company on 25 Feb. 1598, and first printed in quarto later that year".

But how much earlier does the phrase go?

Probably not much further, as the earliest quotation in the Oxford English Dictionary is also by Shakespeare, in Love's Labour's Lost just a few years earlier:

Euer and anon they made a doubt.

The OED has the first known publishing as 1598 and Wikipedia says it is "believed to have been written in the mid-1590s for a performance at the Inns of Court before Queen Elizabeth".

Often people claim Shakespeare was the coiner of this-or-that many thousand words in the English language, but it's often the case that as a writer of many famous plays, it's more likely his use has survived the ages when someone else may have written it earlier and we've lost their text. People may have already been using them in unrecorded speech for decades. His use is often the most well-known, and dictionary compilers liked to include his quotations in dictionaries.

As a good example, I found an eight-year antedating in The Countess of Pembroke's Arcadia (1590, Book II) by Sir Philip Sidney:

But euer and anon turning her muzzell toward me, she threwe such a prospect vpon me, as might well haue giuen a surfet to any weake louers stomacke.

Thank you for this inquiry. It is rich in meaning.

Movable feast was historically liturgical, with Easter, as you pointed out, being its most famous example. Hemingway probably appropriated it in his notes for his heady cultural experiences in Paris, but in fact, it was his wife Mary who named the book which was published posthumously. The story behind it is charming.

It probably started out as a liturgical appellation, but was taken over by the public to mean all the things you mentioned, including progressive dinners. Its recent use has most commonly been in magazines, explaining why it is used with abandon for any purpose desired.

Therefore, you are correct to say it is not an idiom. I would say it was first an appellation for a feast day which has been appropriated by the secular public as a phrase serving a number of possible uses.