The $1800 you paid, in December 2017, is claimed on your 2017 tax return, even though it was for the 1st 2019 term. You may claim a tax credit based on that amount (adjusted for any other 2017 payments and scholarships applied in 2017).
No, your cannot deduct the $1800 from the $3300 you received in 2018.
You are correct, there is no need for her to amend, since her total line 1(form 1040) income is under the $12,000 standard deduction. You are also correct that the $3300 goes on her return, not yours.
Yes, you can deduct the approximately $1000 for books and other supplies you spent from the $3300 she would report as income. But, there's a better way. You should claim it for the education credit, on your return. Furthermore there is a loop hole to claim some of the tuition paid by scholarship (see below).
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There is a tax “loophole” available. The student reports all his scholarship, up to the amount needed to claim the American opportunity credit, as income on his return. That way, the parents (or himself, if he is not a dependent) can claim the tuition credit on their return. They can do this because that much tuition was no longer paid by "tax free" scholarship. You cannot do this if the school’s billing statement specifically shows the scholarships being applied to tuition or if the conditions of the grant are that it be used to pay for qualified expenses.
Using an example: Student has $10,000 in box 5 of the 1098-T and $8000 in box 1. At first glance he/she has $2000 of taxable income and nobody can claim the American opportunity credit. But if she reports $6000 as income on her return, the parents can claim $4000 of qualified expenses on their return.
