Soju

With family and a baby and unemployment can add to uncertainly, but there are reasons why you decided to get your green card . 
With green card, you would definitely want to come here to the USA and live here. There are certain conditions that is required when you have a green card.   You spent so much work and effort in getting green card, don't lose it.  It can be hard to leave friend, family, and job to immigrate to the USA, but I am sure everything will be okay. Many of us here have done the same and are doing great.   You can establish your new life here and opportunities are great here.  I do not know any other place in the world that can offer many opportunities and freedom as here in the USA.  I say embrace your new home and wife's parents.  I am sure they be glad to help you until you can stand up on your own. 
 
 

I am very happy for you about the baby, and sorry to hear about the father.
It seems to me, boy's father was just not ready to be a father.
Since he entered UK on student visa, it's not like it's easy to simply pack up and go back to original country. It seems he is happy where he is now.
Financially, father should support the baby. Maybe father will realize that he may still want to be part of boy's life.

One thing that I may suggest is try marriage counseling . Maybe there is something that is making him keep going back to Ex..
I think solution is to find the root of this behavior..

You can get notarized from many places and not just from the embassy. Depending on what you are trying to get translated and notarized, many law firms in Seoul can do them for you.
If you just want documents translated to English and have it notarized, it will cost you some money. You might want to shop around.
Also try to gather all the documents ready and have all of them translated at the same time so that you don't have to keep going back and forth. If each visit requires notarization it will hit your pocketbook.