Hi i am an acoustic engineer. If you have any question please ask.
1) the wall that is heating up has it had any rockwool treatment? like a 40kg or somthing under 80kg?
Normally when local contractors do soundproofing they use thermal/heat insulation rockwool instead of the high denisty special brands like NoiseBlock DFB.
You would need someone to check this to confirm.
Your problem is not to difficult to fix both for soundproofing and thermal.
If you really do have bad thermal problems and not caused by rockwool, then a multi layer heatproofing membrane will sheild the house.
Please give me some more details for me to give you more advice, if you get really stuck i can come over for free and help you find a solution.