An individual cannot sue the U.S. Government. Agencies or individuals within the U.S. government can be sued but not the United States itself. Anyway, having cleared up that little distinction, didn't the LPR have competent representation at the removal proceedings? Or did he go about it pro se? There is always the appeals route -- BIA and then federal appeals. You did not give any information about whether appeals were done. You should consult an immigration lawyer with criminal alien expertise to advice you on options moving forward.
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I wonder whether he was represented by counsel or not, and whether all of these issues you are discussing now were raised in court, or on appeal. All of this matters because it is important to know whether he exhausted all of his appeal rights before he was deported. Consult with an experienced immigration attorney to find out what he can do at this time.
Contact immigration attorney Gintare Grigaite at 646-407-2331. Answers on AVVO do not constitute legal advice and do not form attorney-client relationship. Always consult an attorney for a legal advice.
It may be possible but there are many loopholes created when suing the government. Contact an attorney ASAP as in certain cases there are statutory deadlines for filing. My firm handles these types of cases.
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Legal disclaimer: The statement above is general in nature, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship.