Hey Nick - hopefully I can answer all your questions.

Is there anything other than her ZED ticket she needs to prove she is my mother
No. The only things she needs are her CX validated ZED ticket and a government issued ID. Airline ID is only required of the airline employee.

Secondly, is it likely that AA staff in ORD would accept an ORDLGA coupon on the one ORDJFK flight there is each day?
AA staff in ORD should accept the ORDLGA coupon for ORDJFK or even ORDEWR, since LGA/JFK/EWR are co-terminals. You may want to have Mom refer the AA agent to N*TVLZEDREROUTE (Sabre reference page) should they give her any trouble.

If I checked in at AZO in the morning and told them I was aiming for the JFK flight, what if I then got to ORD and had to take an LGA flight. My bags would go... missing?
The agent in AZO should check your bag to LGA , if that's what's shown in your PNR. If, however, you are transferred to the JFK flight, the bag could go either way really depending on whether there's time to pull the bag and whether your transfer to JFK was voluntary or because you were not accommodated on the LGA flight.

Or would AZO just tag them to ORD and make me recheck?
AZO should not make you short check to ORD.