You have too many problems.
I am not sure how the traction control gets involved.
Are you sure your battery is fine?
The DPF should regenerate as long as the car moves faster than X speed on Y gear. For the second generation 2.0 engine this is as low 25mph on 3rd gear. Even if you are in traffic and do not move the regeneration keeps happening but takes longer to complete.
When you turn off the engine without the regeneration being completed, the regeneration will start as soon as you restart the engine. You will need to do some driving before it starts.
Read some more posts about the auto braking. It is not an easy to resolve issue. Somebody had the sensors readjusted.
You have scbs, traction control, DPF, soot warnings. If Mazda cannot read any faults, did they ever wonder it is something like the battery causing all these problems? If you went to a generic garage, some of their machines do not read all cars. So you may be unlucky with them.