if its 4x4 it can be the transmission! easy to fix but u will drive only on 2.
you need new tirods
Steering wheel shakes
Tires out of balance, bent wheel, tire defect, or suspension or steering parts worn.
Warped rotors
allignment
could be bad alignment
Your AC should have NOTHING to do with your barking system, as it is not tied into the vacuum of the engine. If your wheel shakes under braking, then you need to replace your rotors and pads.
Tire out of balance or bent wheel.
1 of 2 things or both, alignment or balance.
I have seen this with bad or unbalanced tires.
It could be wheel balance or worn out steering joints. Have it checked by a trusted garage soon!
Could be that the steering wheel rubber bushing is worn or lose. This bushing is located in the engine compartment, between the end of the steering wheel coulomb and the steering wheel likage...Tom