Probably seemed like a good idea at the time. My wife was in car accident once where the airbag deployed. For some reason her immediate reaction was to roll down the window. People in stressful and perilous situations sometimes do things that make no sense.

I don't really know, but it hardly seems like good practise to pick up a bullet. Don't crime scene investigators usually take calibrated pictures of important items such as bullets and often try to determine the trajectory?

There's also the chain of custody to think about - the bullet could easily have been swapped with a different bullet after he left it on the stretcher.

The bullet they are talking about is the "magic bullet" that hit Kennedy and Connally. The bullet that hit Kennedy's head distinegrated and they found fragments.