I am trying to make a foldable quadcopter for a competition. The competition consists of launching a rocket to an altitude of 5000ft, where the payload (or in this case, quadcopter) will be ejected, where 4 arms will fold out of the shell and will need to guide itself to a landing zone.
We are unsure of a few things though. Obviously, we cannot have the quadcopter running while it is inside the rocket, so we plan to use a photo-resistor to tell the quadcopter to start doing its thing (i.e fold arms out, boot and run controller) after ejection but would something like this be feasible to do with a pixhawk controller? I read that most drones require calibration on the ground before taking off, and obviously something like this isn’t realistic for us. Does this also affect us too?
Are there any other potential problems we might face if we do find a controller that can do what we want?
This sounds tricky! 
So PX4 does not do (or require) calibration before each flight meaning you should be able to calibrate it beforehand.
However, the estimator (e.g. EKF2) on PX4 requires some time to initialize, as well as the GPS needs some time to get a 3D lock. Therefore it would take too long to boot PX4 mid-air. Instead, you should be able to initialize everything on ground and then arm PX4 once the arms are unfolded, etc.
This might still be tricky because the estimator might not be tested/tuned for high accelerations that you might encounter. Also, the tuning on 5000 ft might be slightly off due to the thinner air.
Is there any documentation regarding this?
That’s a good point. The payload should be seeing a maximum of 10gs on the flight. Not sure if this would mess anything up with the fc, sensors, or motors.
Yer you’ll have to read up on the docs and get familiar with the platforms, do tests etc. I don’t think I can answer all this for you.
