The real story
In a 2016 study, UC Berkeley biologists filmed American cockroaches squeezing through gaps of about 3 mm by flattening their bodies to a fifth of their normal height, then running while squashed. Their armour can take forces of nearly 900 times their body weight without injury, and the team built a palm-sized robot based on them for search-and-rescue in rubble.
Their family tree is ancient, with a twist: the 'roachoids' of more than 300 million years ago, long before the dinosaurs, were ancestors of both cockroaches and mantises, and true cockroaches came later. As for living without a head: a roach breathes through small holes along its body, not its mouth, so it can last about a week until it dies of thirst.