She can't if she wants to stay in this country.
Yup.
From what I understand, you can't marry anyone who is already married (unless they divorce their current partner or become a widow)
no
divorce him, legally the marriage was illegal since his name was fake its like you were never married
yep. no divorce = still married.
You need a divorce!
no because divorce is only for the couple married
You already have a husband and just because you want another husband unless you divorce your first husband (at least in Western Countries) then you would be a bigamist which is against the law. File for your divorce; get the divorce and you are free to find another husband.
No
If the 'illegal' marriage was registered then yes.
No, it universally illegal to marry if you are already married and until the divorce is final you are legally still married.
If the immigrant married a US citizen, then that's a done deal and she is no longer illegal. It sounds like you need to take care of your own affairs, starting with divorce proceedings.