His current situation;
I doubt the child is a permanent resident based on what you have posted about him here. I think he entered the US with inspection via an official port of entry (he was picked up at an airport and not dropped off by some dodgy coyote) on an
i-94 (a "visa waiver" which allows 90 days of travel within the USA but
waives any right you have to adjust your legal immigration status, as the name implies), and simply overstayed. Since he is a minor he will not have racked up any "unlawful presence", but he will as soon as he hits 18.
If Amy gets off her fat ass to sort his shit out;
Amy will have to have had, and be able to prove, she has physical
and legal custody of the child for at least two years prior to initiating the legal immigration process. To do this she will have to petition a US court for legal adoption of the child. Once this legal custody is obtained and she has had it for two years, Amy will have to go through the process of petitioning for an alien relative (filing form
i-130). For this she will need to provide proof of her US citizenship, proof of her lawful custody and proof of joint residence. At the same time she can file
i-485, which is adjustment of the child's status to one of legal standing (green card). All of this must be done before the child turns 16.
The whole process is very expensive and long-winded. The i-130 and i-485 alone cost $485 and $1140 respectively, this does not count the cost of biometrics, nor the filing costs for the initial legal adoption or lawyers fees, should you opt for one. It also takes up to 14 months for approval on the i-130/i-485 provided you are not hit with a RFE (request for evidence) if you did a sloppy or half-assed job on your paperwork.
I also just plain don't think the first step (obtaining legal custody) will go well given her own admission to being asked to sign a form waiving her rights to child/foster care. I don't know what form it is but with a little more time I will find it. There's also her sketchy red flag marriage and criminal record to consider.
If Amy doesn't get off her fat ass to sort his shit out;
The child will hit 18 years old and begin accruing unlawful presence. No longer a minor and with no legal family in the US to petition for him, he must traverse the hellscape that is US Immigration Law alone. If he wants to obtain legal presence he will have to return to his home country and remain there while he waits for approval of his visa-of-choice through a US consulate. In his case, he will be denied at the interview based on his inadmissibility (accrued unlawful presence), at which point he will have to file
i-212; the application for permission to reapply for admission into the United States. These typically have a really low approval rate but I think this kid would stand a good chance, sob stories tend to get these approved and the hand he's been dealt is shitty.
His other option is to just continue living as an illegal alien and accruing that unlawful presence, and risk being caught and deported to a land he barely knows.