Since cock is American slang for male genitalia, it’s fallen out of use in America (not in England!) except when specifically referring to cock fighting. Chicken is genderless, hen is the female, rooster. cock and cockerel are male, and chick refers to the younglings (of either sex).
What is the difference between “rooster” vs. “cock” and “hen” vs ...
The full Oxford English Dictionary just says the -cock part of ballcock, shuttlecock,... comes from the male bird sense. Which is ALSO the origin for cock = penis (& derivatives like cock-up = shambles) AND cock = stand up straight (as a fishing float, or cocky = overconfident).
Here's what " cock-a-hoop " means - Cock-a-hoop : triumphantly pleased or boastful : happily exulting // Are agents and builders cock-a-hoop that more money is coming into the housing market through building societies and local authorities? — Audrey Powell Basic research showed that it comes from the mid 17th century - : from the phrase set cock a hoop, of unknown origin, apparently denoting ...
The remark would be "well, that was a cock up"... the mistake becoming known as a cock up and giving name to many other accidental happenings. Alternative: Cock up is a well-known nautical expression. The Cock is the upper foremost corner of a gaff sail rigged sail. The Head is the upper edge and the peak the upper after corner.
The NOAD reports that the phrase (chiefly British) cock a snook means "place one's hand so that the thumb touches one's nose and the fingers are spread out, in order to express contempt;" it is also used with the meaning of "openly show contempt or a lack of respect for someone or something." He spent a lifetime cocking a snook at the art world. As for the origin, the dictionary reports it is ...