"Your fly is open" "You mean my flies?"
In Britain the term was always flies, as in your flies are undone.
The only people I have heard refer to a fly in this regard are Americans. However the two expressions can sound the same, and the difference not be apparent, since an American might say your fly's undone which sounds a bit like the British term flies.
I think the British expression goes back to the days before zips came in (1960s?) when they were always buttons - hence suggesting a plural rendering, flies being short for fly buttons.
I'm not old enough to remember anyone saying "there's a star in the east", the usual modern euphemism is to tell the unfortunate he is "flying low".
I think Americans had zips before we did, the British male for a time considering them too effeminate, before they caught on. ("Real men don't wear zips in their clothing" may have been the sentiment.)
The OED entry with examples follows. It draws no specific distinction between British and American usage, but from the examples it is fairly clear which ones are which.
Something attached by the edge. Cf. flap n. 4.
a. A strip or lap on a garment, to contain or cover the button-holes; hence something used to cover or connect (see quot. 1884). spec. (frequently in pl.) the piece of cloth that hides the fastening at the front of a pair of trousers; also, the fastening itself.
1844 Queen's Regulations & Orders Army 154 [Trousers] Open in front, with a Fly and Five Buttons.
1884 E. H. Knight Pract. Dict. Mech. IV. 351/1 Fly, the fore flap of a bootee. A strip of leather which overwraps the front vamp and receives the strings or other fastening.
1941 I. Baird He rides Sky 234 A pair of tennis shorts with zipper fly.
1942 E. Paul Narrow Street i. 6 The professor..turned toward the pissoir, unbuttoning his fly en route.
1952 ‘Vigilans’ Chamber of Horrors 27 The words button one's fly are offensive only to the prurient.
1953 M. Dickens No More Meadows i. 49 Champ, your flies are undone again. That boy! He'll be arrested yet.
1959 R. Fuller Ruined Boys ii. viii. 128 ‘Your flies are undone,’ said Matley primly.
Part 1 of your question: As an American, I can say that I have never heard the term flies in this context before, but it seems that the words are interchangeable as they refer to the same thing regardless of type of fly/flies. There are a number of slang terms for this in America such as "XYZ" (eXamine Your Zipper), "Your barn door's open," "zipper," etc. However, someone might say, "Your fly's open" which would sound similar, but is not the same, as this would translate to "Your fly is open".
In addition, neither I, nor anyone else I know, would make a distinction based on the type of mechanism used to keep your pants closed. Buttons, zipper, Velcro or whatever it is, it would still be called a fly or zipper.
Part 2 of your question: It appears the word "fly" was coined to indicate a tent flap in 1810. As typical in languages, the word was probably associated with the action of an open tent flap and that it flapped or would appear to fly, and bob's your uncle.
In 1844 a book about military clothing made the first recorded use of "fly" to indicate a "pants opening."
As to the origin of flies, it could also be referring to a tent flysheet that had "flies" to tie the flaps closed to the tent pole.
So, perhaps the origins are very similar, but not the same. One origin is referring to the action of the object and the other is referring to the object itself. Which makes perfect sense in how they are used when referring to something being "open" versus "undone".
Well, that's my 2p.
I can only speak to the first question: As an Australian I've only ever heard "your fly's undone" in person. I've heard the BrE "your flies are undone" a few times on TV and it sounds very strange to my ears, but I definitely understood the meaning without undue effort.
I'm generally pretty careful to ensure my fly stays closed, but on the odd occasion someone has had to inform me of my mistake, it's most commonly been "flying low" spoken quietly with a light cough.
Incidentally in Australia we would usually (in my experience) refer to zips as being "undone" or "unzipped", not "open". We don't tend to "close" a zip, we "do it up" or "zip it up" (though it's worth mentioning a zip can be in the state of "closed" in Australian English).
One British English euphemism is 'You've got egg on your chin'.
In most of England it would be 'your fly's open'. I can't quote sources for that, though I do have a lifetime of trouser inattention