Can't answer the second part, but for the first, one of the main problems is documentation.

The Germans had records going back long before Hitler and the Nazis came to power where people had openly acknowledged (on paper) what their religious background was. So when the Nazis came to power, it was quite easy to open the records and go down the list of people who registered as Jewish and round them up.

To get around this would require one to completely assume a new identity, which would have been difficult.

regarding your first question, /u/OldWorldGlory rounded up some previous responses in this post

• How did nazis identify who was Jewish or not?

For the second part, at latest at the Wannseekonferenz, it was very precicely defined who actually is Jewish. It was exactly defined how much "Jewishness" is allowed. I don't have the accurate definition at hand, but just for your imagination (these are taken out my ass at the moment):

We take a given man. His grandparents were catholic, his great grand parents on the male side were both Jewish. This would make him jewish enough for extermination, if only one great grandparent on the female side would be Jewish he wouldn't be considered jewish.

Jewishness was defined with these criterias. If you want the exact definition of "how much was allowed" I can try to look it up later.

Before the Nazis took over the German government, Jews were extremely proud of their religion. Meaning, they would have no problem listing their religion, their children's religion, and their grand children's religion on any sort of document that asked for it.

As far as the second part goes, I couldn't tell you anything about that, but it seems like something interesting to research!