In the opening credits scene, Castle and Alexis are talking, and Alexis's necklace keeps switching back and forth between on top of her shirt and under her shirt.
The car is stopped by a powerful EMP pulse which would burn out all electronics including the listening devices attached to the car so they would not have been working when Becket opened the plastic container and said, "We have what you want."
It is proposed that a metal fragment from an altitude chamber got lodged up Marie's nose. There is no possible way a piece of metal would fly off a million dollar piece of scientific equipment.
Ted Carter comes under suspicion because his key card was used to enter the research facility where the victim worked. When he quit he would have had to return his key card or his access would be revoked in the system rendering the card useless so it would be impossible for him to enter the facility with his card.