I suggest you obtain a transcript of the plea entered. You should contact the Clerk of Court for the procedure in obtaining such a record. In the meantime, contact some local criminal defense attorneys to discover if the plea can be undone.
A plea may be withdrawn for many reasons, but timing is important. Sometimes, the fact that a court appointed lawyer failed to file certain motions may be enough to have the plea withdrawn. It's tough to say given how little information we have to go on here. But, once you have a transcript of the plea itself, at least you can continue your search for a local attorney who may be able to find a defect in the plea.
Also, I would obtain (via a public records request) a copy of the "discovery" in the case from the State Attorney's office. A copy of your friend's file may reveal the fact that important aspects of the case were neglected by the court appointed attorney. Again, that may provide additional grounds for withdrawing the plea. Start now!
It is possible but very difficult, and time is of the essence. If he took a plea deal the judge would have been required to advise him that by pleading guilty or no contest he was subject to deportation if he was not a citizen. If the judge did not include that portion in the plea colloquy, then yes, it may be possible to withdraw the plea and vacate the sentence.
Two things to be concerned about: First, a motion to vacate the sentence and withdraw the plea must be done ASAP. If your friend waits too long then he can't do anything. He should hire an attorney to handle it for him. Second, this doesn't make the case go away. The prosecutor wasn't willing to reduce the charge to misdemeanor before, so there's definitely no guarantee of it happening even if your friend is successful and withdraws the plea.
This is not to be considered legal advice nor does an attorney-client relationship exist.
First things first: court appointed attorneys ARE attorneys. Public defenders are some of the best attorneys and have tremendous amounts of experience. State law will determine whether or not his criminal conviction can be challenged which, in part will be dependent on how long his case has been closed. Your friend and his wife should consult an attorney who can handle both the criminal and immigration issues.
He was likely fully advised on the record about his immigration consequences so it will be very difficult to re-open that case on that issue. Attorneys are all subject to the same rules and duties to their client. It makes little difference how they are being paid, whether private or by the state. They had representation, period. That's what makes it difficult to overturn. There's going to be a transcript somewhere documenting the what the defendant understood.