Just to be on the safe side regarding the back cam, you can try first a hard reboot and if it doesn't help, then a software restore.
But if you ask me, you have for sure 2 unrelated hardware issues:
1/ The audio and power control cable aka power button flat cable aka PMV cable of your iPhone 5 is starting to go bad. Soon it will stop working completely. This is very common for the iPhone 5, so much so that Apple has a program for repairing this issue for free even if your iPhone is more than 1 year old (check the Support section of their website or visit an Apple Store for details and eligibility), provided that your iPhone has not been tempered with before.
You can replace it yourself, but it is a bit of a PIA. Follow this guide. Be extra careful when prying the battery out. Pry only on the outer edge, never on the edge located near the logic board.
2/ The camera is probably defective too, and needs replacing following this guide.
The reasonable thing would be to have Apple replace the PMV cable (if eligible), THEN you replace the camera.
OR you can just go ahead and do both repairs in one go, since if you are replacing the PMV, you would already have gone through all the steps needed to replace the back cam. Look at the 2 guides and you'll see what I mean.
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