1. Yes, this tuition fee paid by the PI is tax free to you.
2. You should enter that it is tax-free employer-provided assistance as the employer is making an "investment" in you and thus they take that as an expense over time (or that is how they should be taking it but that has nothing to do with your taxes).
3. The $657.40 should be entered in as expenses paid out by "you", but also reimbursed by the fund. This used to be pretty straight forward before the SD and ID change as you'd enter it into the "Miscellaneous - Other - Subject to 2% of your AGI" and it would get added to your itemized deductions (anything over 2% of your AGI). Given that this is a complete wash in and out, you should still enter it here as it will go as "informational purposes only" to the IRS, but really has no actual bearing on the bottom line of your taxable income as the amount is so small - and it nets out to zero anyway.
4. No, since there was no cost to you, you would not receive any benefit from these potential credits. Reminder, too, that for the LLC, you should always refrain from using up one of your eligible years if the cost to you is minimal - you'd want to save them for a year where the cost is far greater as these years given do not have to be consecutive.
