"mother kicked the child out of the house" is reason enough for the mother not to claim her.
There are two types of dependents, "Qualifying Children"(QC) and standard ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test. Only a QC qualifies a taxpayer for the Earned Income Credit. If the child did not live with the parent for more than half the year, she cannot be a QC. To be a standard dependent, the parent would have had to provided more than half the child's support and the child must have less than $4200 income.
"In 2020 she will be providing more than half of her living expenses, so she should qualify for the credit next year." Maybe not. A full time unmarried student, under age 24, is only eligible for the refundable portion of the American Opportunity Credit if he supports himself by working (earned income). You cannot be supporting yourself on parental support, 529 plans or student loans & grants. You usually must have actually paid tuition, not had it paid by scholarships & grants. It is usually best if the parent claims that credit.
"I thought the AOTC was limited to 4 years, regardless of whether it was used as a deduction or taken as a credit". You're kinda using "AOTC" as a generic term for tuition tax break. AOTC (American Opportunity Tax Credit), LLC and TFD are specific credits or deductions. Each has it's own rules. Only the AOTC is limited to four times.
"In essence, the stimulus check acts as an advance of your 2020 income tax refund. This means when you prepare your 2020 income tax return, there will be a line to include the section 6428 credit. The credit on your 2020 return is subtracted by any amount received as a stimulus check in 2020. If the amount you received as a stimulus check is less than the credit you are due, the difference will be included as part of your 2020 refund. If you have been overpaid by receiving the stimulus check, however, you will not be required to return any excess amount".
