Oops edited to add: I meant to quote luckygirl.
I don't claim to be an expert but from my own experiences with immigration (I'm a permanent resident with a Green Card) I do know the answers to these questions and will answer if that's ok with the mods, if not, I apologise and please delete.
This applies to any foreign citizen getting arrested for a crime while abroad, your embassy can help with advice and finding a lawyer etc but they cannot interfere with the legal process in a foreign country. LE of the country where the crime was committed have jurisdiction until such time as charges have been answered, the suspect is cleared or convicted and any sentence imposed has been served only then will immigration authorities take jurisdiction and proceed with deportation measures.
As far as AB's immigration status is concerned, he is most definitely NOT an American citizen, he hasn't had time, he's only been here for 2 years and you have to hold a Green Card for 3 before you can apply for citizenship.
If he did it legally, right now he may be a permanent resident with a conditional Green Card valid for 2 years. Within the ninety days before that GC expires AB and EB would have to apply jointly for the conditions to be lifted and a permanent GC issued. Basically they would have to prove they are still living together as man and wife and have been all along, have joint financial commitments - bank accounts, leases etc -, and have filed joint tax returns. Here's the stinger for AB though, even if he were still living as man and wife with EB and he could convince her to sign the joint application, he would also have to declare any arrests and/or convictions on the application, and they don't just take his word for it, the FBI does background checks. In his case, since he does have arrests and outstanding charges against him, that would almost certainly lead to him being denied a permanent Green Card and lead to his deportation.
If he's here illegally, and he IS charged and convicted he will have to serve whatever sentence is imposed HERE before he can be deported. If he's NOT charged, I'm sure LE will inform ICE who will take steps to have him deported if he doesn't leave of his own accord.