A literal translation (Protection Squad) is unlikely to tell you much. Please see the related questions.
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Originally organised by Hermann Goering, the gestapo was Prussia's secret state police. However, it eventually became part of the SS and operated throughout Germany and in occupied areas. Members of the Gestapo generally operated in civilian clothing but, in occupied areas, often wore SS uniform with police-style (rather than SS) rank insignia. Membership of the Gestapo didn't necessarily mean membership of the SS.
The Gespato was the German Secret Police when Adolf Hitler was rounding up Jews for death camps.
There were several different military police. The Army had a military police who were wore the standard army uniform but wore a different eagle and swatika on their uniform and helmet. They wore one on their LEFT sleeve. The SS were the political police and were the ones who wore black uniforms. Before the war, the SA were the police. They wore brown shirts and the small hat (similiar to a kepi) and were called the "Brown shirts". They were replaced by the SS.
Yes, from 1934 Himmler was head of the SS and the entire German police, uniformed and secret. He was head of the entire Nazi terror apparatus.
The SS or Schutzstaffel developed AFTER World War I, so the question is inapt. In World War II, the SS police was a Nazi paramilitary police force.