Checking Grounds is the easy step 1. If all Grounds are good, things get.... fun...
First thing I would do is pull the halfassed work done to the wiring. Get the harness behind the dash as clean as possible. You don't have to get an OEM stereo, just undo all the crap for now. Cap the ends of all the wires you expose to prevent any crossing, aarcing etc.
Check for some of your issues. If they are gone, great, if not then further teardown is needed. Are the HUD and DIC factory options on your car? If yes, great, if no then you need to dig deeper and find any modifications to the wiring. This could involve literally pulling the entire dash out.
Also, not for nothing, check out the BCM. If something has happened to it, it could cause many of these issues.
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