Yes, if a green card was issued during the year, then you are a citizen for the entire year. there is no need to make the first year choice election. it does not need to be amended.
Although, it doesn't specifically address EIC eligibility of children of green card holders, i would suggest following the letter of the law and not claim EIC for this year. This is one of those gray areas that happen and I tend to lean conservative on my positions.
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