Sorry, I should have explained earlier. Many of us don’t comment on the common things that can easily be found by searching on here.
Your post-03/2003 325i has the MS45.1 DME, which will trigger a transmission fault when the MAF sensor is bad or disconnected. That MAF sensor also has a built-in intake air temperature sensor, hence why you have both faults.
The EGS (auto trans control module) requires the calculated load from the DME over the CAN bus and will trigger a transmission fault when the calculated load is wrong or missing. DME requires the MAF sensor value to calculate the load.
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