No, the Nazis were against the socialists. They had the same root word, but that is like saying a calculator is made of calcium, because they have the same Latin root.
In German, it would be Nationalsozialismus, one word. Both socialism and Nationalsozialismus use the Latin socius which means allies, companions, or in these cases a common team. Where it differs is what that common team is for. Nationalsozialismus is where the common team works for the nation in a political/government arena. Socialism is where the common team works together for the economy in the economic realm. Nazism is a system of government, whereas socialism is a system of economics.
Nationalsozialismus being one word meant it was not a type of socialism, but rather a word using socius as a root.
They actually came for the communists first, and then the socialists. They claimed that the Jews were behind both, so came for the Jews right after those two.
Sort of. It is more complex than you make it sound. Auschwitz was first a "quarantine camp". The first prisoners sent there were German common criminals who were prisoners, but also internal guards. The next group of prisoners sent there were Polish intelligentsia. They were not POWs, and almost half of them were Jewish. They were sent there because they were intelligent capable Poles, not because they were Catholic or Jewish. They were mostly not in the military, and were not given any military honors. A "quarantine camp" is where the Germans sent people while they decided what to do with them. Most of them were sent to Germany for killing, but some were released, and few ended up dying in the camp.
The Nazis first started exterminating people with a test group of Soviet POWs. They were used because they were already being held and transported. They were used only as a test for the Jews who were going to be exterminated there. At that point Auschwitz went from a "quarantine camp" to a "concentration camp"(really an extermination camp). Most new arrivals lived for less than an hour. It was originally meant for Jews, but they had no problem adding anyone else who caused them problems, or that they did not want in their future Germany.