There's a Star in the East

Solution 1:

I can't speak to the first question, but as for the second, it's common in the US to say something along the lines of your barn door is open. This assertion is supported by a survey done by the Dictionary of American Regional English that you may find interesting.

Solution 2:

Examples that mention the "star in the East" euphemism are disappointingly thin on the ground – to say the least.

New Bats in Old Belfries, published in 2005, written by Maurice Bowra (1898-1971)

• ‘Showing a star in the East’ was a euphemism for showing an unfastened fly-button, to avert which peril signs warning ‘Please adjust your dress before leaving’ used to adorn public conveniences. Source

Dictionary of Euphemisms (1987) by R. W. Holder

• star in the east (a) an undone fly-button
  An oblique warning from one male to another which seems not to have survived the zip-age

In the chapter dedicated to Human Bodies, page 44, the same author writes

• We males must still adjust our dress after urination or we will find ourselves at half mast, catch a cold, be told that we are wearing a canteen or Turkish medal, learn that Charlie's dead, fly a flag or low, let Johnny out of jail, have a medal showing, hear that it is one o'clock at the waterworks, be warned that the shop door is open, or see a star in the east.

Eric Partridge in his Slang: To-Day and Yesterday (1961) seems to confirm Howard's theory that the origin is derived by the British army.

• A military synonym was star in the East. The usual euphemism is a button showing.

  However, in Shorter Slang Dictionary (1993), Partridge and Beale annotate the following

• star in the East, a a fly-button showing. Public-school slang. Since around 1915