As you mention, the situation is not at all clear. What you need to do about it is completely clear, however.
First, some history. Two resources:
This summary written by a law firm, and
Baylor's web site which reviews the most up to date information.
Interestingly, the recent decision against Mayo "splits" the circuits (something the first article suggested was very unlikely) and theoretically opens a path to the SCOTUS.
Anyway, what to do about it is easy. You can't control FICA payments. Either your program takes them out of your paycheck, or they don't. You can't claim them back on your 1040. In fact, the only one who can get them back is your residency program, and then they can pass them on to you. So, most programs continue to pay FICA and hope for a big fat refund in the future.
As for state taxes, that will differ between the states and there is no one answer. However, the issue above is being argued for the "student exemption" for FICA only, so the answer is probably no.