Ok stop there... a battery that is new does not by any means indicate it state of health. This is why the first order of business is to load test your NEW battery! As for the connection it reads like you only did at the battery and not at the chassis ground and the starter. Contact cleaner is not the proper electrical cleaner in the engine compartment of a MAZDA of your year and generation. Unless the label indicate sensor safe. Both these points require you remove the cable and inspect the area of attachment (mounting) and the cable itself. I pretty much know you did not check the cable at the starter because it is not easy to access.
Opening up any ECU.TCM.PCM etc., without experience serving them and know what each of the specification for the circuits is a sure way for FAIL!

ASE

 

The battery is fine. It starts hot, the headlights are bright on battery and drop the voltage very little. This is not a battery issue and the starter cable is not the issue. I have had a battery in this car in the past that went so bad that the electric power steering would cut out due to voltage drop. The computer still worked fine even then. I have used contact cleaner for over a decade on many cars and electronics and I've never had anything damaged by it. It is very safe to use on connectors.

The car starts and runs just fine except for the codes, hard erratic shifting, traction control not working, and dash lights on. I went ahead and resoldered all of the connector pin joints and all the pins on the surface mount I/C's and it did not fix the issue. Fortunately mazda doesn't try too hard to keep you from servicing these boards there is just a very thin epoxy. All the smd fuses are intact and there are no obvious shorts so I'm hoping it is the BGA chip solder joints as I have seen many many BGA solder joints fail on new electronics that heat cycle a lot. The only think left to do is reflow the BGA chip which I saved for last due to the higher risk. I will do that tonight and see what happens. I have nothing to lose as this car is about to go to the scrap yard due to many other headaches I have had with it through it's life.

Edit: After further playing around with it I noticed the car will die or not start at if you flex the board just a little by the BGA chip. This indicates bad solder joints under the BGA chip. At this point I don't know if I should buy a used ECM and send it off to one of these services to copy the memory from my old ECM. I'm curious what a dealer would charge to pair a new ECM to the car but the only problem is I don't have a Mazda dealer locally.

Edit: I completely removed the Mitsubishi labeled BGA chip and reballed it. This did not fix the issue. The car still runs and drives but I am back to square one. I think it's time for the scrap yard. I'll keep the car sitting around for awhile longer just in case something comes up