In other words say I cooled water based ice cubes down to 250 degrees below 0 on the Celsius scale. Then I plop those ice cubes into a drink. Are they going to last any longer then regular refrigerator ice with respect to how fast they melt ?
Yes.
They need to warm up to 0 before they melt, and it takes long to do this the colder they are.
However, the real genius of ice is the 80 cal/g latent heat of fusion - it’s why ice works so well to keep things cold.
Agreed.
If we make some grossly incorrect assumptions and say that the heat capacity of ice is 40 J/(mol K) at all temperatures, then it will take 1 kJ to warm 1 mol of ice from absolute zero to its melting point. But it will take 6 kJ to turn the ice to water once it reaches the melting point.
I haven’t done this sort of math since high school, so you may want to see if someone corrects an error before you believe me. Lots of rounding was involved.
Of course. Out of curiosity, why would you think they wouldn’t?
Depending on the composition and temperature of your drink to start with, and how much -250 degree ice you dropped in it, your entire drink might freeze solid. 
I’ve read more than once about people claiming that ice is always 0 C (or 32F, whichever). I think the confusion is from not being able to cool the water that the ice is floating in below freezing.
I should have stated question differently. I was really wondering if there would be any really significant delay in melting with the super cooled ice or if the ice would .more or less immediately come to regular ice temperature if put in a drink at room temperature. Ice can last quite a long time as a solid. If regular ice lasts 20 minutes (as a solid) in a drink will super cooled ice last say, 40 minutes, or just a negligible fraction more.
Why not use the real numbers? As noted above, the heat required to melt ice is about 80 cal/g. The specific heat of ice is about 0.5 cal/(g K) just below freezing, but falls off as things get colder (divide those numbers in the last column by 4.18 to get cal/(g K)). So you would have to cool your ice to significantly below -160 C to double the heat it would take to turn it to water. This wouldn’t mean it would last twice a long though. Because the ice (if left exposed) or your drink would initially be colder, it would gain heat more rapidly from the environment.
Would super-cooled ice be more likely to crack as it’s dropped into a drink, thus giving itself more surface area versus an ‘ordinary’ ice cube? Or does that only happen to cubes with pockets of air trapped inside?
It’s been a long time since I’ve experimented with ice in drinks myself.
It will freeze large or small or somewhere inbetween portions of the drink you drop it in.
That may be an advantage, or it may mess up the drink.
If you have Ice floating in water after a given time the ice and water will be the same temperature and that temperature will be the freezing point. Of coures if the water is in a small non thermal cup the water on the outside will be slightly highere.
The numbers are real, but the math was off. 0 K –> 273 K takes ~10 kJ, not 1 kJ. That’s with only 1 sig fig and assuming (incorrectly) that the heat capacity doesn’t change. Since C[sub]p[/sub] drops, as you noted, less heat is actually required to warm the ice. I don’t have a table that goes that low. Of course, the crystal structure can change as you cool it off so there are other phase changes to take into account (although I imagine the enthalpy is small), and some of the weirder structures are metastable at lower pressure once you make them at higher pressure. I don’t think you can get ice IX down to 1 atm. Too bad. It would sink in water 
If you cool ice enough, especially if you add pressure, you are going to get some weird state transitions that can turn it into something like a metal with melting point far above normal boiling. It’s possible that some of this stuff might exist on Saturn or Jupiter.