Soldato
Lithium battery. Plus the items will be wraped in foam to insulate and a heatpack to try and help boost temps. Along with the box thats insulated.
It's been done lots and works.
The camera is a push to make switch so the output can fire across that.
If i said im going to 100,000ft you'll have a small idea whats being planned here!
Bit of a bump here but have a look at a site called battery university.
Li-ion cells actually prefer cooler temperatures so you may not need as much foam as you think. I had some rather toasty 18650's today after overcharging them, they actually now perform better after being in the fridge for a few hours, believe it or not!
You still want to prevent condensation however, I recommend looking at that goo stuff that those crazy over lockers that use liquid nitrogen to cool their CPUs use.
Finally, the Arduino code is pretty easy if you've ever done any coding before, you can use a sleep command that adds a delay in milliseconds, i use it when switching relays as they're not instant on / off.
Finally, good luck, whatever you're doing it sounds like a challenge