A: The government has a copy in what's call an "alien file". If your husband ever needs a copy, he can request it from the government through a FOIA request (Freedom of Information Act). If the file was for you, I would tell you to hang on to it. But since it's for your ex, you can either give it to him if you like, or you can shred it and tell him to file a FOIA if he ever needs a copy of his file. Your choice.
A: If they let you back in the US, you should be fine as long as you don't depart again for a while. And by "a while", I would recommend you not depart for AT LEAST a year, if not longer. In order for them to revoke your permanent residence WHILE YOU ARE IN THE US, they have to take you to immigration court, which is a lot of money for the government. It's completely different if you are seeking re-entry to the US. Yes, technically if you are in possession of a facially-valid green card at a point of entry, you can demand an opportunity to see an immigration judge, but in that scenario they can take you to a detention facility while you wait to see a judge and hold you for months. Most people in that situation simply give up and voluntarily relinquish the card. Totally different if you're present in the US pursuant to a lawful entry. Then they can't hold you in detention. Bottom line, you're fine. Go ahead and petition for your husband. But do not depart the country again for AT LEAST a year, preferably longer.
A: 194 countries use the Interpol database that will tell them about convictions in any country that reports to Interpol. It is normal and common for immigration officers in those 194 countries to run your name through the Interpol system when you pass through the immigration post at the airport.