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There are no passport validity requirements when entering the country that the passport belongs to. In your kid's case, travelling to the US with a valid US passport (no matter how close to the expiration date) is perfectly okay. The airline should give you no trouble with that.
According to travel.state.gov for India:
PASSPORT VALIDITY:
Must be valid for six months beyond date of visa application to obtain a visa.
According to Transiting Through Singapore:
As an illustration, if travellers intend to depart from Country X, transit in Singapore, and fly to destination Country Y, they are strongly advised to check the requirements of Country Y.