Short answers, pulled from the story.
The Italian Social Republic was officially declared on the 23rd of September 1943. Mussolini announced the creation of the Republican Fascist Party three days earlier and proclaimed the resumption of war alongside Germany and Japan shortly after.
SS-Obersturmbannführer Otto Skorzeny led Operation Eiche to free Mussolini on the 12th of September 1943. German commandos used gliders and paratroopers to locate him at the Hotel Campo Imperatore before flying him to Bavaria.
The de facto capital became Salò, a small town on Lake Garda. The state existed for nineteen months before total collapse in May 1945 with its forces bottled up at Collecchio.
Eight thousand Italian Jews died during the twenty months of the Salò regime. Most killings occurred under the administration of the Italian Social Republic while Black Brigades committed atrocities against political enemies and resistance fighters.
RSI Minister of Defense Rodolfo Graziani surrendered what remained of the state on the 1st of May 1945. This surrender took effect one day after German forces capitulated following an unconditional instrument signed on the 29th of April.