In Spain, we have the following idioms, which I think are similar in meaning to the English ones mentioned in the OP.
For “The cat is out of the bag”, we can have "estirar de la manta" (to pull the blanket off). This means, however, to deliberately reveal some secret that someone else doesn’t want revealed. Acting like a whistleblower, in other words.
“El contable amenazó al presidente del banco diciendo que, si le detenían, estiraría de la manta” (“The accountant threatened the president of the bank by saying that, if the police were to arrest him, he would reveal everything he knows”)
For “Cat got your tongue?” we have "¿Te ha comido la lengua el gato?" (“Has the cat eaten your tongue?”). It means the same as in English (i.e., being tongue-tied).
“Hijo, háblame, díme qué ha pasado… ¿te ha comido la lengua el gato, acaso?” (“Son, talk to me, tell me what happened… The cat got your tongue, or what?”).
For “When the cat is out, the mice will play” we have "Cuando el gato no está, los ratones hacen fiesta" (“When the cat is not there, the mice will have a party”). Same meaning as in English.
“El jefe se fue a mediodía, y ya sabes… Cuando el gato no está, los ratones hacen fiesta” (“The boss left at noon, and you know… when the cat is out, the mice will play”).