Ok, I have absolutely no idea what is going on with your poor axie. I'm going to go out on a limb here and say he is really, really sick. In my honest opinion, I think you should get him to a vet with experience with amphibians - heck, to any competent vet at all, at this point - today.
In any case, cool means 18°C or below. If you can manage that in some corner of your house, you don't need to fridge him. Fridging him will not hurt, though, as long as the temp in the fridge is above 5°C.
If you need something to distract you while you are waiting for your Dad, you could measure the water parameters in the aquarium (ammonia, nitrite, nitrate, temperature) and post them here. You could try again to make some pictures to post. Not really sure what else to recommend.
-Eva