Short answers, pulled from the story.
Slavkov u Brna is located about 16 kilometres east of Brno, in Vyškov District in the South Moravian Region of the Czech Republic. It sits on the border of three geomorphological regions and the Litava River flows through the town.
The German name Austerlitz evolved from the Latin Nova sedes, meaning new settlement, passing through forms including Novosedeliz (1237), Nausedlicz (1322), and Nausterlitz (1460) before settling as Austerlitz by 1611. The Battle of Austerlitz, fought near the town on the 2nd of December 1805, took its name from the town.
After the Battle of Austerlitz, an armistice was signed between France and Austria on the 2nd of December 1805 in the historic salon of Slavkov Castle. The castle was originally a Renaissance residence built at the end of the 16th century and was rebuilt in Baroque form by Italian architect Domenico Martinelli in the 1680s.
Italian architect Domenico Martinelli redesigned Slavkov Castle into its current Baroque form in the 1680s. He also designed the Chapel of Saint Urban on Urban Hill, built in 1712, which was badly damaged during the Battle of Austerlitz and rebuilt between 1858 and 1861.
Fred Astaire was born Fred Austerlitz, and Austerlitz is a Jewish family name whose bearers trace their ultimate family origin to Slavkov u Brna. A Jewish ghetto existed near the town's commandery from the 14th century, and a synagogue built in 1858 and a Jewish cemetery still remain from that community.
Paris named the Gare d'Austerlitz, the Pont d'Austerlitz, and the Quai d'Austerlitz after the battle. A village in the Netherlands was founded in commemoration of the battle, as was a small town called Austerlitz in New York state.