As far as your first question, you can read the Nazi's complaints against the Jews in any number of books, such as "My Struggle" by Hitler.
Regarding the second question, the identification of "non-practicing" Jews, let me first of all say that jwentings answer that they were measuring their foreheads is just nuts and has no relationship to reality.
In old Germany, everyone was heavily documented and had to carry around a book with them that listed everything about them including their places of residence, their occupation and place of employment, their religion and their parentage, so it was well-known whether a particular person was Jewish or not just from their passport. If you did not have a passport, you automatically got arrested. The criteria for being a "Jew" was if any one of your 4 grandparents was Jewish. If your passport said you (and your grandparents) were baptized Christian, then as far as the authorities were concerned, you were Christian.
After 1937 (I believe, correct if wrong) the Nazis began stamping passports with a red "J" right on the first page if the person was Jewish. This simplified things for police who did not need to go leafing through the passport to determine if the person was Jewish or not.
Just to give you an idea of how anal and totalitarian they were about it, you generally needed at least three kinds of documents: the passport, basically containing your life history, a residency permit and a work permit. The residency permit described you physically and was needed to prove you had a right to live in a place. So, for example, if you had a residency permit for Coburg and were stopped in Hamburg, they would ask "What the hell are you doing here?" and you better have a good answer and documentation to back your answer up. The work permit (called an "arbeitsbuch") listed your employment history and any important details (like getting fired).
Below is a page in an "Ahnenpass", meaning an ancestor addendum to a passport listing your ancestors:
In cases of someone without papers, they would try to figure out who you were, and generally would imprison you until they knew who you were. In cases of foreigners, like Russians, they would rely on the locals to tell them who was Jewish or who was not. Also, in Russia, the Jews tended to exist in specific communities, so the Germans would assume you were Jewish if you lived in such a community and would assume you were Christian otherwise. In the case of a foreigner captured abroad, like on a road without papers, if it was a male they would see if the person was circumcised and that would be the deciding factor. If it was a female, alone and with no males (a highly unusual case), it would be at the discretion of the arresting officer to decide where to send her, but in most cases the Germans did not bother arresting random refugee females without a specific reason.
There is a funny story about Alekhine, the famous chess player. He tried to cross the militarized Polish-German border in 1937 without papers (!!!) When the police questioned him, he said "I am Alexander Alekhine, Chess Champion of the World. This is my cat. Her name is chess. I require no papers." Of course, they arrested him.