That was my exact question. I checked teanglann.ie and found that there was a lot of similarity between Ulster "comhrá" and Munster "comhartha". In every dialect there were distinct differences between them. I wish we could know which dialect we were hearing before we answer. This would solve a lot of confusion.
It's useful for hearsay or faulty memory. "They said there was a sign at the door." So you go and you look around for it. And bear in mind that this is only the most literal English translation for an Irish phrase. Sometimes getting used to the Irish thought pattern is useful.
Not really. Some notices can be considered advertisements ("Pub Quiz next Friday in aid of XYZ"), some clearly aren't advertisements ("This office will be closed on Monday"), but "Fógra" is used in both of these cases.
"Léigh sibh an fógra aréir" - https://www.duolingo.com/comment/16732811
"Chuir siad an fógra ar an mballa" - https://www.duolingo.com/comment/11038355
"Thug an dlíodóir an fógra do mo mháthair" - https://www.duolingo.com/comment/5932212
No, it doesn't. If the sign is stuck to the door, you can say that it is "on the door" in English and ar an doras in Irish. If the sign is on the wall next to the door, or on an easel in front of the door, it is not "on the door", it is "at the door", and you say ag an doras.
A sign that is "on the door" is, logically, "at the door", but a sign can be "at the door" without being "on the door". The sign in this exercise is ag an doras - "at the door". You can't conclude that it is "on the door", any more than you can conclude that it is on the moon. It might well be on the door, but the Irish sentence doesn't say that, and it explicitly avoids using the Irish construction that means "on the door".