Education

The first place to look is in IRS publication 970 which can be downloaded from https://www.irs.gov/pub/irs-pdf/p970.pdf .  (You can also order a free printed copy through the IRS website https://www.irs.gov/forms-pubs/forms-and-publications-by-us-mail )  To address your questions:

1) You can always not claim a child as a dependent.  I did that a few times with my college age children in order to get a tax benefit.

2) Your child's standard deduction depends upon whether she can be claimed as a dependent, not whether she is claimed.  So, yes, she needs to check that box.

3)  Pub 970 , page 13 has a flowchart to determine if a person is eligible to claim the credit.  It shows that you cannot claim the credit for a child not shown as a dependent on your return, but the child can claim the credit.  This is summarized on page 20.  Page 21 also states that she cannot use the 40% refundable portion of the credit to reduce her taxes below zero.

 

However, any money that was used by the 529 plan to pay qualified educational expenses excludes those expenses from the educational credit.  The good thing, though, is that 529 plan money can be used to cover room and board and some other nonqualified educational expenses.  Page 16-17 of that publication show examples of legitimate expense shuffling that, for example, may allow scholarships to be treated as taxable income and thereby increase the out-of-pocket payments of qualified expenses such as tuition.