Alas, the refrigerator is dead.
That smoke was the refrigerant escaping. Theoretically, you can patch the hole or holes and recharge it. But that will cost more to get right than buying a new one.
Next time you need to defrost, turn it off and let the ice melt. This will take an hour or two at no risk to the refrigerator. If you need to do it in minutes instead of hours, use a hair dryer.
Freezers aren't really designed to freeze things, they're designed to keep things frozen. To change one pound of 32°F ice to 31°F ice, requires the removal of .5 BTUs. To change one pound of 32°F water, to 32°F ice, requires 144 BTUs. Therefore, it requires far less energy to keep an item frozen, than it does to freeze the item in the first place.
Almost all sources (including the USDA) agree that a freezer should be kept at 0°F (-18°C), to store frozen foods. Though a temperature of -5 to -10, is recommended to freeze food.
If the back panel of the freezer is popping off, you might want to investigate what is causing it to pop off. If there is ice forming that is causing the panel to pop off, it could mean that the refrigerant in the system is low, and there may be a leak. This is probably a unlikely scenario, though it is possible.
The more likely cause, is poor air flow.
If there's not good air flow through the condenser, the heat removed from the refrigerated compartments will not be removed from the refrigerant. At the same time, if the air flow through the evaporator is poor, heat will not be removed from the refrigerated compartments effectively. Maintaining good air flow within the refrigerated compartments is also important. The cooled air needs to move throughout the compartment, so heat can be taken away.
- Make sure the condenser fan is working, and there's not a lot of dust and debris obstructing the flow of air across the condenser.
- Make sure there's good air flow across the evaporator, and that the evaporator is not icing over (this may require disassembly of part of the refrigerator).
- Make sure there's good air flow within the freezer, and that items are not blocking the vents.
Best Answer
It depends on the fridge, but I doubt it would cause additional problems for the unit.
I wouldn't trust it to keep your food safe, though. A single bag of ice is cold, but won't cool a large space. If the air circulation in your refrigeration compartment is broken, the food more than a foot away from the ice is unlikely to be cooled much. If the air circulation is operating (but blowing warm air), it could overwhelm the cooling effect of the ice.
I'd try to put everything necessary into chest coolers (maybe ask a neighbor for an extra one if necessary), then ice or dry ice from the store to keep it cool overnight until the repairman arrives. Especially if the freezer is working, so you don't have to deal with those items.
My nearby grocery sells dry ice and it has those cheap blown styrofoam coolers on top of the frozen food chillers. I know all this from the last time I had to do a refrigerator repair.