Why does my 2009 Mazda6 seem to intermittently cut out?

your are losing power DC12v to the cluster(loose fuse)
or the communications to it are bad.
check all fuses! cab/bay
and scan the OBD2 16pin DLC port near drivers knee's
scan it. using a scan tool. end guessing for sure COMM errors.
you can not fix modern cars lacking such a tool.
for sure EFI and all things in the car on the network.
USA car, not told. and cars are not the same all over earth. not yet.
seems like the IP cluster is not happy at all.
the radio uses the same Can_bus network ! hummmmmm
aka,. gM lan, or low speed lan
and pin 1 on the scan tool tool.
the cluster is digital and is called IPC, scan the car,
or get it fixed in a real shop with real tools.
the SM is here

https://eautorepair.net/Marketing/Default.asp

Intermittent windows, gauges, radio - Ford Truck Enthusiasts Forums

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Oddly enough something similar recently happened to my 2009 650i recently purchased. The system would intermittently cut out and restart, turning on the radio even with another source selected or radio off. No idea if there is a short or software issue causing it, but it started when I put the navigation DVD in and tried to use it. Took the DVD out and restarted the system, problem hasn’t returned. I’ve read about others having similar issues with the navigation system causing the radio/idrive to cut out as well. It’s a long shot, but worth posting so others don’t spend money chasing a problem solved by using your smartphone instead of the car’s navigation.

Seems to be a common problem:
http://newbeetle.org/forums/car-audio/47902-complete-guide-factory-audio-systems.html
Hope this helps they seem to know their stuff and there is a complete pdf manual for Moonsoon but I would hazard a guess and say you may well need a new one or aftermarket one soon

By Who?
Was the Battery disconnected? in the Process?
Sounds Like a possible short in the Sound contrl Module

intermittent are hard to find.
as they say" if it ain't broke, you can't fix it" applies
but in truth, we lay in traps to find this.
this is true with all machines, not just sx4
facts: (random no starts) but always cranks fast.
only fails cold engine. never hot. (does restarts hot work?)
seems yes.
seems you have full 12vdc power at all times.
seems the engine is ok. as it is not heat rlelated or the classic hot soak fails.
so what is it.
well, its 1 of these.
1: lost spark, simple tests, do so when failed.
2: bad fueling. (too much or too little)
the tests for each failure are dead simple and work on any engine.
ask.

The bluetooth and radio are located in the radio itself. I would suggest having the radio replaced at you local toyota dealership

Sometime the computer in car will shut down a circuit when it sees a problem even if it only occure once. You can try to disconnect the battery for at least 2 min so all sys power off (some have capacitors that mantaine a charge for short time). If a battery disconnect does not clear problem you have go to dealer and have them to clear codes and reset you modules and your preferences

You might have a problem with the body contro module, have it tested for codes.

You may have an intermittent switch or possibly a shorted or broken wire. As for the radio, I'd yank it and put an after market deck in. GM radios aren't known for their quality. But, you could check the back of the radio to see if the antenna lead is plugged in securely.
Hope this helps.