How sensitive is the cockpit voice recorder? Can it pick up sounds from the passenger cabin?
It depends a bit. One of the microphones for the cockpit voice recorder is an "area microphone," which is designed to record whatever someone standing in the cockpit would hear (the others record from crew headsets and possibly the PA system). This area mic is continuously recording; it does not just turn on when there's more than some threshold of noise. If the noise in the cabin is loud enough to have been heard in the cockpit, it will be heard on that mic. If not, it won't.
I only have experience with general aviation (GA) aircraft, that being said, I bet the intercom in a commercial airliner is similar.
There is something in GA called "squelch" which is like a minimum volume limit for the microphone to turn on. This makes it so background noise (such as the engine) doesn't make it into the intercom system. If commercial airlines use this same technology it is very likely that any background noise from passengers is not going to get picked up by the intercom system, and it will not be recorded.
That being said, a commercial airline is much quieter than most GA aircraft, so they might not use squelch. However, the microphones are still going to be pressed up against the pilot's lip, and any sound from the cabin that makes its way through the air, and cockpit door is going to be so quiet that it will be indistinguishable by the time it gets to the pilot's microphone.