I do not know why you are not responding, but you are not charged per question, per response, nor per answer. So after I give you my answer, you can continue to ask me questions without additional charge until you are satisfied. I apologize if it is a site issue that you posted and the post did not go through.
Anyone that is a Lawful Permanent Resident (LPR) and is outside of the U.S. for 180 days or more within any 12 month period (not necessarily calendar year) creates a rebuttable presumption of abandonment of residency. That presumption can be rebutted by evidence to the contrary such as filing U.S. taxes, maintaining a home in the U.S. and paying that mortgage or rent, maintaining a U.S. drivers license, U.S. bank accounts with significant movement, etc.
Someone that has been outside of the U.S. for more than 1 year without first having an approved re-entry permit has abandoned their residency and only in very few exceptions (such as serious illness) can they get it back.
Here is an official link:
So hopefully she applied for a Re-Entry Permit before leaving the U.S. or she could have a problem at the Port of Entry.
As to your child, you would have to apply for a green card for the child which can take a year and a half or so. Could the child come in with a B-2 tourist visa or Visa Waiver? Maybe, but it is risky because those visas are not supposed to be used when there is immigrant intent.
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