I'm interested in exactly how the camps operated differently from each other. Also were people separated based on age/health/capability or was it purely based on religion and ethnicity or perhaps geographical proximity? Additionally why did the Nazis have extermination camps at all? It seems like working people to death is strictly more beneficial for them?
u/commiespaceinvader answered a lot of my questions in this post, but I'm still unsure of how it was decided who went to what camp
Good question! I hope that you read the link given in another answer and that you found some answers there. However, I will try to give you some idea of the situation. In addition, I write about the difference between death camps and work camps here.
As you can see from the link I provided, there was a significant difference between the death camps and the concentration/work camps. In short, concentration camps served a variety of purposes from slave labor to incarceration of criminals or other "undesirables." They could house German criminals, German homosexuals, Poles, clergy who protested the Nazis, communists, etc. The purpose of the concentration camp also varied depending on the commandant or the location. In some camps, the death rate was higher than at others (excluding the end of war transports of Jewish prisoners that greatly inflated death rates at those camps to which they were transferred), while at others, the goal seemed to be economic to some degree. In other words, the camps were much like every aspect of the Nazi regime, varied according to the nature of the leader in charge of it.
In the concentration camps, people might be separated based on any number of reasons, but often Jews were kept separate from others. Kapos (prisoner functionaries) were often common criminals and abused the others, with Jews and homosexuals tending to be on the low end of the hierarchy. Some camps themselves became known for specific types of prisoners (Dachau, for example, was known for the number of clergy it held).
The most famous "selections," however, are associated with Auschwitz and applied to Jews. In these, men and women were separated from one another. If the work camp portion of Auschwitz was full, the entire train might be gassed without any selection whatsoever; however, in instances where "workers" were needed, the weak, elderly, children, and often women with children were selected for gassing. Those who were selected for work were then taken into the work camp.
Regarding your last question, there was a debate between 1939 and 1941 regarding what was to be done about the Jews. As the General gouvernement of Poland became a dumping ground of Jews (creating much consternation on the part of Hans Frank, the Governor-General, who opposed it being used in such a way), the various German rulers of Polish cities with Jewish ghettos each had different goals. In The Origins of the Final Solution, Christopher Browning states "a the core of the dispute over ghetto policy was a split between "attritionists" and "productionists." As the ghettos became longer term features rather than simply collection points, Jews were paid for work and an emphasis might be on economic productivity. Alternately, such as in Lodz, while Jews worked, the focus by the German officials was on getting rid of the ghetto through "attrition."
What changed this policy seems to have been the invasion of the Soviet Union and with it the drastic increase in the number of Jews under German control. Ghettos had never been a permanent solution to the Jewish Question and had been much opposed by almost every Nazi in Poland. Moving the Jews into the East became a fascination of most such leaders. In the Soviet Union itself, Jews became the victims of the Einsatzgruppen as they were tasked with eliminating threats to the rear of the German army. This radicalization of Jewish policy soon encompassed those Jews in occupied areas as well. The Polish ghettos were liquidated and the Jews of Europe were relocated to the East.
All this is to say that the policy of extermination MIGHT not have been the most economically beneficial, although the linked post from the previous comment will give you more information on that. Ultimately, however, economic benefit was not the only principle at play. Enrichment of the SS leaders was key and speed was an important aspect of that. Hence theft of Jewish property trumped long term economics and slave labor. Also, anti-Semitism and a desire to exterminate the Jews played an important role. Finally, the radicalization that came along with the invasion of the Soviet Union provided impetus as well.
Hope this helped.