Yes, your math is correct.
Yes, do NOT enter the 1099-Q, at all.
Yes, for the parent's taxes, just enter $4000 as being box 1 and $0 for box 5 of the main 1098 and then proceed. Don't enter the 2nd 1098-T. It has no effect and just risks errors being made.
Yes, you have the entry method for the student's return correct too. Verify the output before filing. Although you could use a short cut: Enter the main 1098-T with numbers in boxes 1 & 5 that force the answer ($1580 taxable scholarship).
If the $1580 is her only income, technically she does not need to file a tax return. But, you may want to do so, anyway, to document the taxable scholarship, that allows you to claim the full AOTC.
