1. Only the person claiming the individual can claim the education credit. See IRS Q & A about education credits. The 1098-T is reported on the parent's return to claim the education credit. If the student has taxable scholarship income, then it is reported on their return as well.
2. In What is the Kiddie Tax? we see that level of investment income does require the form. See also Topic No. 553 Tax on a Child's Investment and Other Unearned Income (Kiddie Tax). It may be that the $1k wages student does not need to file a return and therefore, instead of filing 8615 on their return, you should file 8814 on yours. See About Form 8615, Tax for Certain Children Who Have Investment Income and About Form 8814, Parent's Election to Report Child's Interest and Dividends.
You don't mention scholarships which are also subject to the kiddie tax and may be part of your difference. They count as earned income for filing requirements but unearned income for the kiddie tax.
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