Code P means taxable in the year immediately prior to the year of the Form 1099-R. TurboTax's text in the selection box assumes that you are entering a 2021 Form 1099-R, but you are entering a 2022 Form 1099-R.
1. Her contribution for 2022 is unaffected. No need to do anything with regard to 2022.
2. She would receive a return of $1,250 of her 2021 contribution with the distribution adjusted for earnings. What she does with the money is up to her, it's just cash at that point. This particular distribution would not be eligible for rollover but she can use her cash to fund the rollover of the March 4 distribution (the only distribution eligible for rollover) or she could use the cash to fund part of her 2022 contribution if she has not already contributed the full $6,000 for 2022.
3. Vanguard would know it's a rollover contribution because you would tell them it's a rollover of a regular Roth IRA distribution that was made on March 4. Nothing needs to be mentioned about its original intent to be a correction of the 2020 excess contribution. Mentioning that would only potentially confuse them.
Regarding the investment gain, I had used the $1,000 figure that you had mentioned in conjunction with the return of the $3,398, but perhaps you just mentioned $1,000 as an example an not the actual gains attributable to the $3,398 returned.
Regarding the second set of questions:
1. The deadline for rolling over the March 4 regular $1,250 distribution is May 3, 2022. As long as she files her tax return timely or submits a filing extension request (Form 4868), the deadline for obtaining a return of contribution is October 17, 2022 (but it makes sense to do it by the regular filing deadline, April 18, 2022).
2. Leaving things as they are won't affect the amount that she is eligible to contribute to an IRA for 2022.
Disregard anything I said about the Retirement Savings Contributions Credit. If you claim her on your tax return or she is a full-time student, both of which seem almost certain, she does not qualify for this credit.
