The bottom line is fairly simple (despite the confusing wording).
The children cannot claim themselves and the stimulus/rebate credit because they qualify as the mother's Qualifying Child (QC) dependent (they live with her, they don't support themselves and they are Fulltime students under 24).
The father cannot claim them, as a qualifying relative, because they are the QC of another taxpayer (the mother). He cannot claim them as QC because they don't live with him and the special rules for divorced & separated parents no longer apply, due to emancipation.
The mother is allowed to forego the dependent, to allow the children to claim the tuition credit, but not the refundable portion.
@RK06 said " When creating a tax return for them within turbo tax, my dependents can select "someone else can claim me on their tax return" and "not one will claim me in 2020", and it gives them the American Opportunity Credit. If I didn't claim them, are they allowed to do this?
Yes & no. As previously stated they can claim the non refundable portion. There is a question in the personal info section that asks them if they provided more than half their own support with earned income. When they answer no, TurboTax will not give them the (up to) $1000 refundable portion.
