Where and when was the key invented?

Very probably in ancient Egypt, 6,000 years ago. The oldest surviving key to have been discovered by archaeologists was made of wood and was found in the ruins of the Assyrian city of Nineveh. It was used to lock a door in the palace of King Sargon who lived around 2,300 BC, so over 4,000 years ago.
"Theodore of Samos in the 6th century BC." ~ Wikipedia
The inventor of the key is unknown, as the device dates back to a time prior to historical records being kept. There is evidence that the Egyptians were using a type of key over 4000 years ago.