If you are using original casing then i think there is problem in your phone. When you open your battery cover you will see a small angular point or switch-like mechanism on the right side of battery slot. When you slide your phones’ camera cover, the cover push the switch inward which makes the camera auto on. I think there is problem.
I will like to give some solutions for it:
1. If you are afraid of going to mechanic then you should, first restore factory settings of your phone. Press *#7370# at default or stand-by skin. Careful your all contacts, messages & all data of phone memory will be deleted so it better be you backup them. ( this is the easiest way rather then opening phone, going to mechanic or care center.)
2. Open the phone caring and wipe the camera switch electric circuit position. ( if you are not expart enough then you should not try it.)
3. Go to an expart person he who knows much about it.
4. Though care center take much money, their service is good enough then any other options.
5. In your post you mentioned some keys are not working properly, it could be if you are using a third party casing. Then i will suggest you to use original casing.
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have a good time, as-salamualaikum.