A ri is a Japanese unit of measurement that is about 3.93 kilometers, or 2.44 miles. It is defined as the distance a person can walk in an hour at a reasonable pace.
"A Stick Of Time", the episode title, refers to incense clocks, which originated in Song Dynasty China and spread from there to Japan and Korea. They measure the passage of time: minutes, hours, and so on and so forth, by measuring the time it takes to burn sticks of incense. Case in point, Toranaga-sama lit and extinguished the incense stick to mark the beginning and end of his meeting with Madame Gin, of "tea house" fame. Even the use of the incense to measure time was a relatively new practice for the era.
In the podcast for this episode, due to the wide divergence in age between Toranaga and Saeki, it is theorized that they are half-brothers, since it was a common practice in Japan for a man of high status to have multiple wives called consorts and a child's status was based on the seniority of the consort. In the source novel, Toranaga and Saeki were indeed half-brothers, but they shared a mother. Toranaga's father died when he was a boy, and his mother remarried a lesser lord.
Madame Gin was advocating for the creation of a special or red-light district, called Yoshiwara in the city of Edo, which would later be renamed as Tokyo. It would become a collection of tea houses and sort of licensed, controlled, protected and taxed by Edo shogunate, Edo government. Gin sees underutilized value in her courtesans, petitioning Toranaga for the creation of a guild in his new empire. This formalization of the profession would lead to the creation of the highly organized and hierarchical Geisha class, and be the foundation of a rich cultural district in Edo, built on the skills of entertainers who brought their performance and artistry to the new city.
Toranaga threw a stone at a wooden archway. This archway, made of two vertical posts connected by two posts on top. is actually known as a Torii Gate, which demarcates the sacred and the profane, or the world of the spirits and the world of man, with respect to the Shinto faith. The gates act as a passageway into a shrine's sacred space. When passing through a torii gate, it is customary to walk to the side instead of straight down the center. It is believed that the center space is reserved only for the passage of kami (gods). In some regions, local custom dictate by throwing stones at torii gates, and placing them on top, is believed to bring you luck. This custom exists in Yamaguchi (western tip on the main island of Honshu) prefecture among other, but not in the Kansai (major city of Osaka) or Kanto regions (Toranaga's domain).