The name "Particicution" is a play on the name "Participaction", a government corporation founded by Pierre Trudeau in 1972 in an attempt to get Canadians into fitness. The corporation lasted until 2000 in Margaret Atwood's homeland of Canada.
In the book, the story takes place in Cambridge, Massachusetts, predominantly in Harvard Square, on the Harvard campus, and in the general area. Harvard Square is a terribly busy place, however, and it would have been almost impossible to clear the Square for the scenes in question. In addition, Harvard University has a "no filming" policy that prohibits any filming from taking place on their campus. North Carolina substituted.
In the original source novel, the main character is known only by her patronymic, "Offred" (or "of Fred," since she "belongs" to a Commander named Fred); her real name is never revealed, though many readers believe it may be June, based on various subtle hints in the text. However, the filmmakers chose Kate as her pre-Gileadean name, and choose to state it clearly.
The repressive theocratic regime that has taken over the US in this movie (and its source novel) is called "The Republic of Gilead." Gilead is a place (or maybe several places) mentioned repeatedly in the Bible (first in Genesis 31:23), both as a geographic location and the source of a figurative or literal "balm" (curative or healing substance). Based on those constant Bible references, there is a well-known spiritual, "There is a Balm in Gilead," that is in the hymnals of many Christian denominations, and in the book The Handmaid's Tale, Offred remembers the hymn and makes a joke about it.