Mazda ain't gonna do much about... they might only refer the
matter to the towing company, which i believe yes, they are liable
for the towing damages.
From your description, it seems that your mazda has a bodykit or
lowered suspension? stock cars usually have high enough clearance
to accomodate the tow thingys... as for the exhaust, was the car
lifted way high up? Perhaps in the future, if you have concerns
about damage to the car while towing, you might wanna ask for a
flatbed tow truck instead.
