Short answers, pulled from the story.
The word petticoat came from the Middle English pety cote or pety coote, meaning "a small coat." In the 14th century, the garment was worn by both men and women as an undercoat, and it was not originally underwear but a visible layer worn with an open gown.
Men in Western fashion wore a petticoat equivalent from the mid-15th century through roughly the 17th century. It functioned as an under-doublet, usually padded or quilted, worn under a shirt for warmth in cooler months. That garment later evolved into what became known as the waistcoat.
Christian Dior revived the petticoat in 1947 as part of his full-skirted "New Look" collection. Tiered, ruffled, and stiffened petticoats became essential to achieving the silhouette and remained extremely popular through the 1950s and 1960s, with some stores still stocking them as late as 1970.
The Japanese equivalent is the nagajuban, worn under the kimono. It resembles a shorter kimono, is commonly made of white silk, and typically has a half-collar called a han'eri attached for protection and decoration. A second underlayer, the hadajuban, may also be worn beneath the nagajuban.
Sybil Connolly's first international collection, held in New York in 1953, was inspired by a traditional red flannel petticoat she saw worn by a woman in Connemara. She bought fabric from a local shop and made it into a quilted evening skirt that was a major success. One of those skirts is now held at The Hunt Museum.
Petticoat government refers to women exercising authority over government or domestic affairs. The phrase appeared in Henry Fielding's plays, an Irish pamphlet in 1780, Washington Irving's Rip Van Winkle in 1819, and novels by Frances Trollope in 1850 and Emma Orczy in 1911. G. K. Chesterton used it approvingly in What's Wrong With the World in 1910.