not following all your questions but will answer what I can and how I can,
#1) what do you mean by the 'basis' - are you referring to a box on the 1099Q? the AOTC has nothing to do with the 1099Q. If you are filing joint and your income is less than $160,000, you are eligible for AOTC. TT should calculate that automatically (assuming you answered all the questions correctly)
#2) did you add in things like Room and Board, non-required books? if these 'qualified higher educational' expenses exceed the Box 1 distribution, then the Box 2 earning are NOT taxable. I think the taxable earnings show up on Schedule 1.
#3) that goes on your child's tax return - not yours. That is why you don't see it. But it is a separate question whether your child is required to file in any event. if their earning income (and this money would be considered earned income) is less than the lessor of earned income plus $350 or $12,000 AND there are no withholding from working, there is no requirement for the child to file a return,
