"(1) Given what TurboTax's message say the beneficiary must report her income, can I pay that in my return, or must I create a new return for my daughter?"
No, you can't claim that income on your return. Yes, the student would need to file a return and claim that income if their income is at or over the filing requirement.
The income from a distribution is claimed by the taxpayer that was issued the 1099-Q. If you were listed as issued the 1099-Q, you report it, if it was issued to the student, the student reports it.
"(2) Does my daughter need to file her own return anyway because adding the two box 2 "Earnings" amount above, it goes above $12,550? "
It depends on how much she needs to claim. It sounds like she may need to claim only 4,741, however distribution income is considered "Unearned income" so 4,741 unearned income would mean she would be required to file.
"(3) Reading some answers, it sounds like sometimes 1099Q do not need to be entered. But what situations should we entered them then? I want to pay the right amount of taxes. Like the other guy, I really do not want to be audited."
The education section is complicated, partly because you have options in allocating expenses and payments. The TurboTax program will try to get you a credit even if that means having the student claim income to free-up expenses for a credit.
Please look over IRS Pub 970 for great tips and examples.
The problem with the student claiming distributions is that it can trigger kiddie tax, in which case claiming a scholarship might prove to be the better choice.
https://www.irs.gov/pub/irs-pdf/p970.pdf
@EarnY
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