Consider, play the victim [card]

victim playing (also known as playing the victim or self-victimization) is the fabrication of victimhood for a variety of reasons such as to justify abuse of others, to manipulate others, a coping strategy or attention seeking. Where a person is known for regular victim playing, the person may be referred to as a professional victim.

Manipulators often play the victim role ("poor me") by portraying themselves as victims of circumstances or someone else's behavior in order to gain pity or sympathy or to evoke compassion and thereby get something from another. Caring and conscientious people cannot stand to see anyone suffering, and the manipulator often finds it easy and rewarding to play on sympathy to get cooperation.

Wikipedia

An expression that overlaps (but not totally) with your metaphor is: To be a wolf in sheep's clothing.

Wikipedia has:

Wolf in sheep's clothing is an idiom of Biblical origin. It is used of those playing a role contrary to their real character, with whom contact is dangerous.

PhraseFinder makes the definition clearer:

Someone who hides malicious intent under the guise of kindliness.

As does McGraw-Hill Dictionary of American Idioms and Phrasal Verbs:

a dangerous person pretending to be harmless.

For the non-violently predatory meaning (pretending to be poor/ weak/ ill/ hapless), you might consider a pity me pitch. As used here on Facebook. The expression pity me is a compound adjective (used attributively); here is another example of its use, from Debra Kidd on WordPress:

So why on earth did I let some negativity on twitter bring me down this week? This post is not intended to be a ‘pity me’ post, but rather an exploration of why it is that a small number of privately educated men seem to think it’s alright to personally attack women they deem to be strong.

Note that she thinks it not standard enough yet to be used without scare quotes.

Two expressions specific to

1. pretending to be poor or using poverty as an excuse, and
2. pretending to be ill or unwell,

are 'poor-mouth' and 'malinger'. These both have noun and verb forms and, in the case of 'malinger', an adjective form.

poor-mouth, n.
colloq. (orig. and chiefly U.S.).
a. to cry (also play, talk, etc.) poor-mouth: to plead poverty.
b. Used (chiefly attrib.) with reference to (esp. unjustified) protestations of poverty or personal misfortune.

["poor-mouth, n.". OED Online. March 2016. Oxford University Press. http://www.oed.com/view/Entry/237577?rskey=d6fl9j&result=1&isAdvanced=false (accessed April 05, 2016).]

An example of attributive use is

Let's talk about what's important as billionaire club owners roll out their poor-mouth propaganda and millionaire players set in motion the mechanism to strike.
(From Daily News, 2002, "Enjoy Baseball While You Can", by Kevin Modesti.)

'Poor-mouthing' may also be used as the adjectival form of 'poor-mouth'.

malinger, v.
intr. To pretend or exaggerate illness in order to escape duty or work; to feign or produce physical or psychological symptoms to obtain financial compensation or other reward. (Originally used of soldiers and sailors.)

["malinger, v.". OED Online. March 2016. Oxford University Press. http://www.oed.com/view/Entry/112944?redirectedFrom=malinger (accessed April 05, 2016).]

The noun forms of 'malinger' include 'malingerer', 'malingering' and 'malingery'. 'Malingering' is also used as an adjective.

Examples of forms of 'malinger' in use:

Rogers (1997) criticized the use in DSM-IV of uncooperativeness and treatment noncompliance as criteria for malingering because those who malinger often are highly cooperative and voluntarily seek treatment, though they do become uncooperative under more direct questioning.

[James Richter (2014), "Assessment of Malingered Psychosis in Mental Health Counseling". Journal of Mental Health Counseling: July 2014, Vol. 36, No. 3, pp. 208-227. Emphasis mine.]

We always called this 'playing the role', sometimes shortened to PTR. It's synonymous with the phrase 'playing the part' that Joel mentioned in his answer.

Oh don't mind her, she's just playing the role so you'll feel sorry for her.

We also used 'world-cupping it', referring to how a soccer/football player who had been fouled would throw himself on the ground, yelling and holding his shin or whatever to exaggerate the injury and make sure the referees would notice. It's the same thing as 'playing it up' mentioned in Joel's answer.

I'm pretty sure he's just world-cupping it and it's not as bad as it seems.

I looked for references and apparently neither of these are well-known outside of my circle of friends and family in the US Midwest. I can't remember when exactly I started using the phrases, but it was more the 20 years ago. With the recent developments with FIFA, "world-cupping it" may have different connotations today.

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.

Here are some more close analogues:

• Playing the part = as in, they've got a role to fill that benefits them, so they won't ever drop character
• Playing dumb = just like the above, but the part is of someone who is too dumb to even understand the argument or deal with.
• Playing it up = like an injury, mental or physical. something happened, and the person has an exaggerated response.
• Faking it = they're putting on an act. might not be malevolent, though.

The above don't necessarily mean ill will (that would be established by context).

• Two-faced = The person acts one way towards you (the good face) and then stabs you behind your back when you're not around (the bad face).
• Harbors ill will = A common idiom for "watch out for that guy"

Note: Playing possum only applies if you pretend you're dead. It's a generalization of:

• Playing Dead = Lying there to escape notice or to get out of something.

You mentioned kids trying to avoid punishment, which kind of sounds like Crocodile Tears. Which refers to fake crying, or pretending to be more sad than you actually are in order to get sympathy or leniency.

for example: "You can knock it off with those crocodile tears, I know you aren't actually sorry."

An interesting possibility is sandbagger. From Merriam-Webster:

sandbag, v.
tr. to conceal or misrepresent one's true position, potential, or intent especially in order to take advantage of

The dictionary goes on to explicate:

In the 19th century, the verb sandbag began to be used to describe the act of bludgeoning someone with a small, sand-filled bag - a tactic employed by ruffians, usually as a prelude to robbing their victims. The verb went on to develop metaphorical extensions, such as "to coerce by crude means." By the 1940s, it was being used of a strategy in which a poker player with a good hand bets weakly, in order to draw other players into holding on to their hands and raising the bet. The use of sandbag has since evolved to refer to a general strategy of playing down one's position in order to gain some sort of advantage.

Rope-a-dope

is the first one to comes to mind

" a boxing tactic of pretending to be trapped against the ropes, goading an opponent to throw tiring, ineffective punches. "

In a single word they're called a weasel: "a dishonest person who cannot be trusted" (MW). The appropriate (slang) idiom is to weasel your way: "to improperly make one's way into." (OSD)

All of your examples are trying to weasel their way in or out of something. Be it disciplinary action, your wallet, your good graces, or doing their job.

"Don't trust him, and don't be fooled by her either. They're both just trying to weasel their way out of this."

How about: " the lady doth protest too much, methinks " .....

This is a well known Shakespearean quotation, meaning that you are feigning distress or ignorance of an issue, too loudly and too innocently, meaning that the more you profess you have no knowlege of a particular issue, the greater the probability that you do. In other words, you are lying about an issue, or that your claim to lack of involvement, is belied by your actual involvement in an issue. Duplicitous behavior.

Taken from Hamlet ACT III, Sc.2

The "wounded gazelle gambit" seems like the closest to what you're looking for. This is named for when a mother gazelle will pretend to be injured in order to lure a predator away from her young and then bound away once a safe distance is reached.

I think the short answer to your question is: no. :-) There's not a single English phrase that conveys the full breadth of the Persian saying.

That said, the existing answers hit most of the highlights, but no one has yet mentioned being "a sleeper".

This is typically used in the context of playing games for money. A sleeper would loose the first several rounds/hands purpose to lure the other players into upping the stakes. Once the stakes are satisfyingly high, the sleeper starts playing at his/her true capacity and wins back everything he has lost and a lot more besides.

Rather than an idiom, English has a word that means exactly what you want

dissemble

"Conceal one's true motives/feelings"

although this can also imply actively lying to avoid detection. An informal phrase that emphasizes the passivity implied in your description is "keep mum".

Definitely not the canonical answer, but before we had play Possum, we had "sham Abraham." A little backstory ...

You've probably heard the term bedlam meaning chaos or madness. Well that word was derived from Bethlehem, which came from an actual insane asylum, the Hospital of St. Mary of Bethlehem in London.

The Abraham Ward of the asylum housed the milder cases - many of these patients weren't even mentally ill, just developmentally disabled - and as a fundraiser each year, the hospital would send these patients out on to the streets to ask for donations. They became colloquially known as Abraham-men.

Never ones to pass up an opportunity, some of the dodgier element of London would dress up in patient clothing, pretend to be mentally ill, and also go around door to door and stand on street corners asking for donations. Hence the term sham Abraham.

In British English idiom there is the expression 'Butter Would Not Melt... (In The Mouth). In C16th it was used to mean a genuinely innocent person but from the C19th onwards it is used ironically to mean that the person is pretending to be more innocent than any truly innocent person ever could be.

Behind this expression is the ancient belief in 'humours'. There were supposed to be four humours in the body and health and moral status reflected the balance of humours. A passionate, angry person with too much of the humour-fluid called 'choler' would be 'hot' and butter would melt in their mouth but an innocent, chaste, gentle, meek person with well-balanced humours (or an excess of the cold humour-fluid called 'phlegm') would be especially cool and butter would not melt in their mouth because of the very low body temperature. If you suspect a child is pretending to be the innocent party in a playground fight where in reality the child was either just as naughty as the other party, or even naughtier, then you would say 'butter would not melt in his mouth...' meaning he is pretending to be more innocent than any child ever really could be.

This expression is less likely to be used about a beggar pretending to be more deserving of charity than they really are - the C16th expression was he/she is a 'sturdy beggar' ('sturdy' meaning fit and well and could work for a living but pretends to be ill to get charity instead). Educated people might still use this expression but self-consciously as an allusion to historical usage. A more commonplace expression in British English would be to say that the person feigning illness or innocence to get favoured treatment is 'having it on'. In direct speech you'd say 'you are having me on' or 'I am not having it on'.

I am not sure, but I think the roots of this expression are also historical. The allusions to wearing 'it' seem to be refering to the very ancient practice of wearing horns or bells as signs of (1) being a joker, trickster, cheat, or (2) of having been the victim of a joker, trickster, cheat. Thus court jesters wore hats with horns and bells to show they were jokers and tricksters. Men who had been cheated by their adulterous wives were called 'cuckolds' and were said to have been given horns by their wives. So, saying 'you are having it on' means you are (wearing invisible horns and bells and) trying to trick me. 'I am not having it on' means I am not going to (wear invisible horns and) be your dupe.

To say the equivalent of 'playing mouse' a British person would have to say 'come off it, you are having me on, with your butter-would-not-melt routine'. Persian is much more succinct!

Maybe: "he's pulling the wool over your eyes"? Applies to most (possibly not all) of the OP's examples.

(Obligatory reference to the most famous malapropism of this phrase.)

IMO using the expression 'playing innocent' can also sound rather formal.

A more emotive, robust or colloquial way of using the expression 'playing innocent', during a conversation is: ‘Don’t come the innocent with me’, This is used for telling someone angrily not to pretend that they have particular feelings or are in a particular situation, because you do not believe them.

Don’t come the innocent with me! https://www.macmillandictionary.com/dictionary/british/don-t-come-the-something-with-me

From Longman Dictionary of Contemporary English: Don’t come the innocent/victim/helpless male etc with me.

British English used to tell someone not to pretend that they are something they are not in order to get sympathy or help from you