Last name change on green card due to divorce

#1
I have few questions about last name change on green card due to divorce, but I will explain the situation first:

1. I am a green card holder (DV2) who divorced in home country and legally changed last name (from husbands to maiden).
2. I have a reentry permit issued with husbands last name, green card with husbands last name and passport with maiden name (because of legal last name change due to divorce).
3. I am currently in home country and planning to go to USA to get new reentry permit for next 2 years.

My questions are:

1. Am I gonna face any problem at the airport showing GC and reentry permit with one last name and passport with another? I will bring divorce certificate translated in english, is it enough?
2. Do I NEED to change last name on green card due to divorce or I CAN keep it?
3. If I reopen a bank account in USA or if I apply for job or when I apply for another reentry permit is it okay if I show up with divorce certificate proving that I am the same person?

Thanks a lot, you all are awesome and helpful! :)
 
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