Sherlock-style situation puzzles

A man is driving his car while listening to the radio when suddenly the radio goes silent. He then decides to leave the road and throw himself off the top of a cliff.

Sherlock Holmes was walking down the corridor. Suddenly he heard a woman screaming, “Please, John, don’t kill me!” A shot rang. Holmes burst into the room. In one corner lay a dead woman and a gun. In another corner stood a postman, a lawyer, and an accountant. Sherlock looked at them, approached the postman, and grabbed him: “I will arrest you for killing this woman.” It was really the postman who killed this woman, but how could Holmes know it?

A mother-in-law disliked her son-in-law so much that she wanted to poison him. But the son-in-law was careful and always ate only what his mother-in-law ate. Over dinner one night, the mother-in-law cut a piece of meat in half, ate one half herself, and gave the other to her son-in-law. After that he died. How did she manage to do it?

Sherlock Holmes had a secret room, and the key to it was in a chest of drawers. He didn’t let anyone in there – neither Dr. Watson, nor Mrs. Hudson. But when Sherlock Holmes and Dr. Watson left, Mrs. Hudson decided to see what was in the room, took the key, opened the room, and entered. She walked around the room, approached a table, opened a book lying on it. Suddenly, she heard footsteps downstairs, and Holmes’s voice, “I forgot something.” Mrs. Hudson quickly closed the book, put it back, went to the door, turned off the light, left the room, closed the door, and managed to put the key back. Holmes took the key, opened the door, looked into the room, and immediately asked, “Mrs. Hudson, why did you enter my room?” How did Holmes know that Mrs. Hudson had entered his room?

A person called the police and told them to ask about John K. Upon arrival at John’s house, the police found him dead. Examining the corpse, the police discovered severe injuries. But the apartment was in perfect order, without any signs of a struggle or break-in.

Following criminals’ trail, a police inspector entered a chess club. Assessing the situation, he told his assistants, “Detain those two!” How did he guess?