Any English equivalent for the Persian idiom "to play dead like a mouse"?

Solution 1:

I'm not exactly sure that the idiom of playing dead like a rat translates to being cunning, but playing dead/pretending to be asleep to avoid injury or confrontation has an equivalent in playing possum.

Your request sounds more like "playing [the] innocent":

"Forgive me, I do not understand," said Cicero, playing the innocent. - NYT
John is playing innocent, and he knows more than he is telling us.

Even more common is playing dumb:

To pretend that one has no or little knowledge (of something); to act ignorant or uninformed (about something).

Another is "dumb like a fox". Foxes are clever foes. When one is pretending not to be clever, this can be used. Also used, in a similar situation, is "crazy like a fox." But these imply hidden cleverness as a disguise, and @Edwin Ashworth's answer is a more common idiom for when really disguising yourself when you have predatory intent.

Solution 2:

In American English, we might use the phrasal verb to play possum. The opossum uses the same sort of defense mechanism as the rats or mice you describe: pretending to be dead or unconscious in order to avoid danger.

This expression has a narrower focus than your Persian idiom, though. We would use it for someone who is pretending to be ignorant or unaware, or physically incapacitated (asleep, dead, etc.); it also means exhibiting this behavior specifically to avoid some sort of danger or trouble. This phrase could be used in your first and third example, but not the second.