"Dean Winchester is as straight as an arrow. He’s a ladies man of epic proportions: the king of the one night stand, the messiah of the friends with benefits paradigm, the emperor of perpetual bachelorhood. Except, apparently, when it comes to Castiel."
NO. Both animals are in a state of 'tonic immobility' or 'playing dead which is a FEAR response and NOT a cute, relaxed or cuddly state. This is damaging to prey animals and not something that should be done.