Can a fridge that runs too cold actually start a fire?

I've fixed a lot of fridges over the years and I've never seen or heard of one catching fire because it ran too cold. The temperature itself isn't what creates heat. If anything, a fridge that's too cold usally means the compressor is running constantly, and constant running wears parts out faster, but that's a reliability issue, not a fire issue. Fires that do happen are almost always electrical: a failed start relay on the compressor, a burnt plug, damaged cord, stuff like that. The temp setting has nothing to do with it.

That said, running too cold is a symptom of something wrong, just not a dangerous something. Most likely candidates are a thermostat that's stuck or drifted out of calibration, a broken air damper stuck open, or a bad thermistor depending on what kind of controls you have. A manual dial fridge drifting cold is often just the cold control failing. On a newer one with a digital panel it's more often the damper or a sensor.

My advice would be to keep it running. Unplugging it wastes the whole fridge full, and frozen lettuce isn't a hazard, it's just annoying. Put a cheap appliance thermometer on a middle shelf and check it a couple times a day. You want roughly 35 to 38 degrees. If it's sitting near freezing, that lines up with your symptoms. While you're at it, feel the sides or back near the compressor. Warm is normal, hot enought you can't hold your hand there for more than a few seconds is not, and that would be the one thing that changes my answer. Also check the door gaskets. A torn or loose seal makes the fridge run wierd in all kinds of directions and it's a two minute check. If the gasket pulls a dollar bill out without resistance anywhere, that's part of your problem. So tell your wife and your brother in law the groceries are safe, get a tech out when you can, and if you smell anything burnt or hear the relay clicking on and off rapidly, that's when you pull the plug.

Same thing happened to me a few summers back, my old top-freezer started slushying everything in the produce drawer. Turned out to be the cold control drifting, like the guy above said. I put a thermometer in there and it was sitting at 31 degrees, about 8 lower than it should have been. Swapped the thermostat myself for maybe 25 bucks in parts and it's been steady ever since.

One more thing worth checking before you call anyone out, if there's dust packed into the coils underneath or behind the unit, clean those off. A dirty coil makes the compressor run forever and can throw everything off. Mine had what looked like a grey carpet under there, vacuumed it out and it ran noticeably quieter that same day.

I had the same frozen lettuce thing turn out to be a damper stuck open on my old side-by-side, food in the back near the vents took the worst of it. You can sometimes hear the damper flap open and shut when you swap the door from open to closed. Mine went quiet and never moved again. One thing I'd add about the placement, that thermometer reading can swing a few degrees depending where it sits. Mine read fine on the middle shelf but way colder near the back wall, wich is exactly where the milk was. Move it around before you panic about the number. Also crack the fridge open once a day while you're waiting on a tech, deep freezes on the fridge side mean stuff in the door bins will still be okay even if it dips below freezing overnight.

First post here so sorry if this is a dumb question, but does the freezer side matter in this situation? Ours had a cold spell in the fridge section a while back and at the same time the ice cream went soft, like the two were fighting each other. Never did figure out why. If the thermostat is drifting like people are saying, would that show up in both compartments or just the one?

Update: took the thermometer advice and stuck one in a glass of water overnight. Read well below where it should be, so the wife was right that something was off. Repair guy came out and said the cold control had drifted, same as the second poster described. Swapped it and the lettuce is crisp again. Glad we didn't unplug it for weeks, the fire thing really had her spooked.