So the camera works perfect outside the housing.
The camera does not work in the housing, but you state the buttons work perfect. How did you determine the buttons work perfect? The buttons may move, but if they are not activating the camera, they are not working.
I'm going with you have a camera housing issue. Until you cycle the buttons with the camera out of the housing, verify they work (you already did) then put it in the housing and try it again. You can press the buttons out of the water, it should do exactly what it does without the housing. Just a little more effort from the button seals and springs.
As for GoPro batteries and cold, I've seen plenty of snowboarding and snow mobile videos of these cameras running just fine in below freezing conditions. May not have the run time due to the cold, but they still work and have lots of usable run time. 50° water isn't going to phase them at all. You would probably need lab conditions to actually see any reduction in battery life.