This is from my quick search:
"A creature with immunities takes no damage from listed sources. Immunities can also apply to afflictions, conditions, spells (based on school, level, or save type), and other effects. A creature that is immune does not suffer from these effects, or any secondary effects that are triggered due to an immune effect."
Bonus question: if you cast icy prison on a monster that is so hot it does fire damage to anyone hitting/grappling does the ice itself take fire damage every round? I would say yes and it takes full damage instead of half.
RAW, no, because the Ice is not hitting the creature with a natural attack/Unarmed strike or grappling it.Technically, most auras specifiy Creatures and so would not damage the ice either.
That being said, having the ice take faire damage each round in those cases is a perfectly reasnoble call for a GM to make, and would probobly be RAI.