We recommend omitting “from themselves” and editing as follows:
Despite being able to talk, infants fail to realize that other people may perceive and know different things.

From our perspective, whether to use a reflexive pronoun might well be determined by the clarity that is needed. In other words, if it is clear that “he” is the only person being referred to in the context, the sentence could potentially be written as “Closing the door behind him, he laughs.”

If however the scene or context includes more than one possible antecedent for “he” (e.g., there are two men near each other), the reflexive pronoun may be needed to specify which of the two is closing the door.

He laughs, closing the door behind him/himself.

If I put the clause in the front, it sounds even more as if the sentence needs the reflexive pronoun.

Closing the door behind himself, he laughs.

If the above example does need a reflexive pronoun, if there an example of a time a dependent -ing clause would not get a reflexive pronoun?

As the first rule on this page states, a reflexive pronoun refers back to another word in the sentence. The reflexive pronoun themselves refers back to the noun cheaters.

Cheaters should be ashamed of _____.
a. himself b. him
c. themselves d. them
What is its grammatical rule about this item?

Yes, the sentence is correct because the pronoun reflects the action of the sentence and refers back to the sentence subject, thus making the pronoun reflexive.

“Not herself” is an example of how idioms can sometimes override grammatical accuracy. Herself is a reflexive prounoun that serves as a predicate nominative in your example sentence.