Sunshine it is so much fun isnt NOT at those early hours for the tantrums....
I have a 2 and a bit year old and he can chuck the biggest wobblies .. not that often but when he does he lets rip!
The cold rooms business: obviously your child is in a bed? (mine still in cot) Could it be a bit of a habit as well? One night she did it and got away with it so she started a pattern? or does she come in obviously cold and teeth shattering etc etc. Im dreading giving my dss 2yr a big bed when this new bub comes as it will be fun and games NOT. And im not a pretty mum when im sleep deprived! So i hear your frustration but all i could suggest is the blanket oil heater and maybe a low gate to keep her in her room ... or use your split system and put it on a timer .... i know it is expensive but more sleep with out weigh the bigger bill
good luck