Q. Should I put it back to Roth (is that allowed)?
A. You have to put it back within 60 days of taking it out. That is a "roll over". You are only allowed one roll over per year.
Q. I can use the conversion part with no tax and penalty for education, and I can use the earning of Roth IRA with no penalty for education but with income tax?
A. Yes. There is a 10% penalty for taking money out of a Roth account (before age 59.5) within 5 years of a Roth conversion. That penalty is waived if the money is used for education.
Q. How should I deal with the $5,250 employer reimbursement (probably in 2021 spring)?
A. You have a choice (actually a tax loop hole). You can claim nothing on your 2020 tax return, since it will be reimbursed in 2021. Or, you can claim a tuition credit, for 2010, because you are out-of-pocket. But, then you would have to treat the (normally tax free) reimbursement as taxable income in 2021. The tuition credit is worth 20%. If you're in the 12% tax bracket, or lower, you come out ahead
Your unlikely to be able to put the money back within 60 days. The money would be better off left in the Roth account, earning tax free money than to be used as short financing. You would, probably, be better off getting a short term loan while you wait for the reimbursement. Check with the school's financial aid office, which may even allow a payment plan.
