The German View of "Jew" was not exclusively that of religion, but of race. So even if the "Jew" in the Nazi sense had converted to Christianity, it was of no matter to the Nazis.

http://en.wikipedia.org/wiki/Conversion_of_Jews_to_Catholicism_during_the_Holocaust

And while I do not have a source available, my professor in my Holocaust class said that there were cases of Christian Priests that were taken and sent to camps because their parents or grandparents had been converted Jews.

Some did, but remember that the process of Jewish extermination was an incremental one. Also remember that Poland was a modern country with extensive record keeping, it was non-trivial to adopt a fake identity with a fake family tree. The identification and ghetto-ization of the jews occurred at a time when many jews did not necessarily see it as an existential threat, so they complied.

Also consider all of the things which tie someone to a Jewish identity: their name, their place of residence, their family and friends, their business associates, their job, their legal documentation. Someone willing and able to sidestep all of those things, with the means to do so, and the forethought to be afraid of persecution would be just as likely to simply emigrate out of the country entirely.

The information in who was Jewish was readily available, and was usually marked by important, unhideable details like circumcision, accent, languages spoken, school, and personal connections.

Additional question for the experts: I get the impression that people were less mobile on a generational basis back then. Would it have been less possible to hide one's ethnic or religious background in those days because one's family had been in a given community for a long time?

I think it's one of those concepts that modern people have a difficult time grasping. I could easily pack all of my possessions and move 1,000 miles away without my neighbors at either end of the move noticing. I doubt many of my neighbors have known my religion, let alone my name. But from reading literature in that era it seems that a person was either from a small town where everyone would know a family's history or they were from a larger city where privacy wasn't that much more abundant due to open windows and the recent transition to urban life.

Also, a Holocaust survivor visited my school when I was a kid. A student asked him how it was possible to identify him as a Jewish person. He said they could look in his pants. Was checking for circumcision a thing?

The Jews also had neighbours. The German occupying forces offered rewards for those turning in their Jewish neighbours. So unless you had your neighbours cooperation you were in huge danger of being caught.