Nothing is known about how Rostock came to be, though its many airbases in central Europe and the fact that it shares its name with a German city (as does Lautern) may imply it is German in origin. Since well before the events of the series Rostock has been at war with Lautern, under supervision of the European Confederation and the United Nations with a set of rules established at some point by an unnamed treaty.
Besides fighting an eternal war, Rostock seems to have developed the Sanka line of fighter aircraft and the Wolfram flying fortress. Rostock is also implied to have created the Kildren, which would mean they have a considerable genetics division. Rostock is also involved in the entertainment industry televising some of their larger battles with Lautern. Rostock seems to be funded privately by investors and sponsors from around the world that occasionally visit Rostock facilities.
the Rostock leadership seems to meet in a larger underground facility in Europe. during the events of Innocent Aces Rostock leadership seems to tightly control operations at the squadron level, by the time of the movie however they seem to have allowed for greater tactical freedom though still occasionally organize major operations involving many units. Kayaba is implied to have been promoted to the Rostock leadership at the end of Innocent Aces, During the film Kusanagi accuses them of intentionally not warning her of a Lautern air raid.
Rostock employs a large military though it is somewhat smaller then Lautern's. Prior to Innocent Aces Rostock suffered a Losing streak to Lautern but do to the efforts of Lynx and the deployment of Kildren, Rostock started to gain the upper hand. however the uprising of Tochika Mozume and several high profile defections including many of the original Kildren and top ace Teacher has once more put the war in a stalemate. Rostock's air force is the scource of its power but it also has a Small army and a large fleet including aircraft carriers blockading the United Kingdom. Mozume was Deputy Chief of Intelligence until his defection which means Rostock has an intelligence Division.