'I've heard every airbag is specifically coded to a certain car, so the code on the new airbag not matching the car is throwing the code.'
No. Module yes, but the bags interchange all day long as long as you use correct year for year to avoid change issues. The module codes to the car because of the writing done to it at accident time. Module only checks the bags for proper resistance of the propellant/ignitor, proving they are in place, and nothing more.
'The new airbag never was set off.'
If the bag did not detonate it could simply be bad. No way you can tell that later except by crashing it again. Or........
Remove the bag from the electrical connector and insert a 2 ohm resistor in place of it at the connector to turn key on, if light goes out the bag is bad. DO NOT TRY PUTTING ANY POWER OF ANY SORT INTO THE BAG ITSELF!!!!!!!!!!
Could simply be a bad clockspring from all the fumbling around the area, I note the OP mentions not hearing the horn relay any more either at the same time, those two things spell bad clockspring.
As I said before, the steering cover can be removed from the bag itself to swap them.