k1 visa greencard holding wife returning for reunion with our China born US Citizen baby question

Hello, our baby was born in China in Nov 2012. My wife's K1 was issued after that. The baby's CRBA/Citizenship was done in Shanghai in Dec. of 2012 and was given a U.S. Passport. The both arrived in the U.S. in February.

My Wife doesn't have her citizenship yet or a U.S. passport, but is a greencard holder now. We attempted to get a visa for our baby as she holds a U.S. passport, but we were turned down as the Chinese want to issue a entry permit.

q1.) Is my wife free to jump on a plane and go to China with just her passport and able to return to the U.S. with just her Greencard?

q2.) It appears that my family must go to the Chinese consulate in NY in person to get the entry permit for my daughter (at least). I don't see any forms that we can fill out to take with us. Does anyone have an idea of how that process works or the timeline?