There is a special rule for divorced parents. From what you describe, the father is the custodial parent. As long as he doesn't try to claim her, it is OK for you to do so. If she is your dependent, you can claim the very generous tuition credit (up to $2500). You cannot claim the Earned Income Credit (EIC), based on her as qualifying child, because she didn't live with you enough.
As others have said, she cannot claim herself. Also, students under age 24 are not eligible for the refundable portion of the tuition credit.
The FASFA eligibility question can't be answered in this tax forum.
Either parent can claim her. Either would be eligible for the tuition credit, but only the father would be eligible for the EIC. It may be worthwhile, preparing returns both ways to see which way comes out best. You can use this tool: https://turbotax.intuit.com/tax-tools/calculators/taxcaster/?s=1
