Three big things come to mind:
1. Have the battery load tested, if it is weak and voltage dips too low while cranking it may cause the IPC to lose its flash memory.
2. Test for a power drain. Most GM's and other late models have a Power Management Mode that will turn off accessories or other circuits if the BCM times out, a door trigger is set, or if the voltage drops too low.
3. There may be an issue with the BCM, IPC, or Class 2 data bus. I doubt it however, I'd put my money on the first 2 tests showing something.
I think the trip odo, gauge reset, and radio are all interrelated. Most likely from a weak battery or a parasitic draw (possibly from an amp, radio, alarm, or many other circuits staying powered up.) If this is a stock radio, the theft lock system in the radio can trigger if the power dips too low or is removed, many newer GM vehicles have the radio programmed to match the VIN stored on the BCM and PCM. It gets this info on the Class 2 or E&C data bus. The IPC stores gauge sweep and trip odo info on flash memory, and it too will lose this if power is lost. Even if it is an aftermarket radio, test the battery and current draw.