It is very unlikely that the ice cracked the cold plate, because the ice builds up from the plate surface outward as water condenses on it and freezes. In order to generate any force on the plate you would need to start with a volume of liquid water which could freeze and expand. There is no place on the plate that would hold a puddle of water big enough to do this.
I can imagine a scenario involving melting, refreezing, and trapped water that could theoretically put pressure on the plate but it seems about as likely as a small meteorite crashing into the fridge. Which come to think of it sounds like the best thing that could happen to it.
Really, the only way a cold plate or freezer coil ever gets physically damaged is by the user attempting to physically remove the ice buildup with a pointy tool.
If the cold plate did spring a leak, the particular dangers depend on the type of refrigerant, but practically speaking, you are not likely to be badly harmed by a leaking home refrigerator, no matter how cheap it is.
Quote from a worksafe site http://www2.worksafebc.com/i/posters/2013/ws_13_10.html:
"Some refrigerants, such as ammonia, are very toxic. They are defined as toxic process gases under Part 6 of the Occupational Health and Safety Regulation. Other refrigerants, known under the trade name Freon, are non-toxic. These include refrigerants such as R-22 and R-410A."
The good news is: your fridge is noisy so it must have a compressor, so it probably uses R-410A instead of ammonia. The bad news is: your fridge is noisy so it is probably about to become TU very soon.
If your fridge leaked at the cold plate, and if the refrigerant was toxic, and if some of the toxic refrigerant was trapped in the ice buildup, the ice might become toxic. You don't need to worry about this because since (a) R-410A is not toxic, and (b) you are not stupid enough to eat the ice buildup from a cheap refrigerator.
To answer the question you didn't quite ask, the thing is likely just completely worn out.
If the box has been slowly getting warmer then ice buildup could be the cause. You could try unplugging it and letting the ice melt (takes maybe 12 hours) and then plugging it in again to see if the cold plate even gets cold anymore.
But if the failure to cool was sudden, then the plumbing has sprung a leak or the compressor has packed in. Or both. What often happens to a refrigerator is this. The circulating fluid contains not only refrigerant but also lubricant for the compressor. A slow leak reduces the pressure in the system until very little heat is pumped, and very little fluid is circulated. The compressor is forced to work longer and harder with less lubrication and eventually burns out its seals or just breaks.
My rule for deciding to fix or discard is this. If the failure was characterized by a long slow decline followed by a complete failure, junk the fridge and replace it. If it failed suddenly without warning, turn it off so it doesn't tear itself up. then try to fix it.
One sign that a freezer is not keeping a constant temperature is bad freezer burn (ice crystals forming around your food, dehydration, damage).
You could use an indoor/outdoor thermometer that records high/low in order to debug this. Each day, record the high/low then reset the thermometer.
If you are seeing a big range in temperature, or, your freezer is getting warm enough to start melting, then I would replace the thermostat.
Best Answer
Ice build up on the coils limits heat that can be removed, causing the pump to run longer, in addition to the colder temperature requirement. That much is known. This is speculation: The pump running longer possibly caused the motor to overheat.
Electric motors of this size usually have thermal overload protection, they simply shut themselves off if they get too hot.
It's best to adjust temperature in small increments, verifying with a thermometer, and waiting overnight to allow the new setting to stabilize before making another adjustment.