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ANSWER: He needs to get "certified copies of court records" from the clerk of the court in which his case was being handled.
As far as the immigration case, you would get married. File the I-130 petition. Then he would make a request with the government attorney at the immigration court to jointly reopen his removal case. If there is no basis that prohibits him from staying, he would ask the judge to consider that he is now married to a US citizen and with the approved I-130, consider reversing or cancelling his removal and adjusting his status or sending the case to the USCIS for adjustment of status based on marriage to a US citizen.
I highly recommend working with an attorney who has reviewed his records and his criminal case to see if it is possible.
I hope that answers your questions.
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