I see an issue with your wiring... your gate pin needs pull down resistor. Not sure about your 12V power source issue though... that's different. Does it happen if you use a 9V source on VCC??? Is it possible your regulator is shot?
I suppose it's possible, but I wouldn't know. Why are these things rated up to 20v or so if they can't handle it? I am able to get the setup working, my question is mostly why a power source would prevent the arduino from running the sketch.
Automotive 12V supply is not suitable here - the strips want 12V at all times, not 13.8V sometimes
and less at others (I assume the 12V is standard lead acid with charging from alternator).
It's currently plugged into the house, so I assume there's an inverter buried somewhere doing the conversion. Throwing a multimeter only shows a voltage variance of about +/- 0.1v. But I don't have enough experience to tell you whether that's a significant amount.
As I mentioned, I am able to get the LED's to do the thing like I want, but only when I DON'T connect the camper wiring to my arduino. That is, the lights work if I provide 5v to power the nano while pushing 12v through the MOSFET to control the strips.
My question is: Why can't this same 12v power supply also run the Arduino? I understand that it's best practice not to because of the heat buildup, but it occurs to me there's something going on the beneath the surface and I want to fully wrap my head around the mechanics of why the sketch won't run.
If the reset pin is less than a volt relative to ground, give or take, the unit is being held in reset and is why blink does not run.
Further, I hope you’re using a proper meter for your measurements, not some bargain tester gadget that can damage a microprocessor.
I'm not quite sure I understand the reset pin. My understanding is that putting a voltage to it resets the arduino. In my tests, I found continuity exists between RST and GND if I use the existing camper 12v wires to VIN, but continuity does not exist between the same RST and GND pins when I supply the arduino with any other power source, including 12v from an alternate AC-DC adapter. So it seems the Arduino is perhaps resetting itself in response to some property of the camper's 12v wiring? But as I mentioned above, it's only bouncing .1v at a time...