If a spell has an effect when it drops someone to 0 HP, it is explicitly mentioned in its description, like disintegrate:
If this damage reduces the target to 0 hit points, it is disintegrated.
Cone of cold does not outright kill a target reduced to 0 HP. The additional effect only happens if the damage outright kills a creature. In the case of a PC this means either taking massive damage that instantly kills them or auto-failing their last death saving throw because of taking damage (PHB 197).
"Until it thaws" just indicates that this is not magical ice that will remain regardless of the circumstances and will melt as natural ice does. You just need to apply heat, e.g. by leaving it in the sun or judicious application of prestidigitation. They become a "frozen statue", not an "ice sculpture", thus after melting out you get a wet corpse that can be raised normally. I personally would also rule that if you use a high level spell that would restore missing organs, like true resurrection, you can use it directly on the frozen corpse.