Of course you can marry him. And there is absolutely no time period you need to wait until you get married. Will this help him immigrate to the US? That is another question.
Chances are he will require an I-601 waiver if he has been unlawfully present. And he will require an I-212 waiver to overcome the deportation. There could be further complications if he ever unlawfully reentered the US after having already been unlawfully present.
If he only entered the US one time, chances...
Agree with colleagues. You can transfer to company B if company B petitions you for H-1B. If duties are "completely different," you will definitely need a new labor cert. Even if the duties were not "completely different," you would need a new labor cert.
I agree with most of the answers already posted. Your son would have to file an affidavit of support which does create obligations if his wife were to accept public assistance.
Regarding this point:
"......that she would have to pay out of pocket for college i.e., no student loans in U.S. and anyone sponsoring her such as our family would also have to prove we can pay for college and support her for five years."
Agree with my colleagues. You should have counsel. It's always a good idea to attach the judge's decision. One reason for this is that it contains (in most cases) your correct A number. If you make a typographcial error, including a copy of the decision could prevent a problem.
Most important thing is filing before the deadline. Notice of appeal must be recieved within 30 days. The due date should be on the judge's decision.
More information is needed. What was he convicted of? How did he enter the United States?
When did he enter the United States? There may be a way to solve this without his leaving the United States. The key word is may.
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For I-601 waivers, the applicant must demonstrate hardship to spouse or parent. That said, that said the hardship to the child can be brought out in the application.
Adjudicators may consider it indirectly. Although legally, they do not have to.