I think you need to take it to Apple and have them see the problem and run diagnostics on it. It almost sounds like someone is using a spy tool (highly illegal) on your phone. It's very unlikely, but that's how it sounds

At this time I am having two issues with my iphone: 1.) The screen goes completely black as soon as I dial a number. Nothing is blocking the proximity sensor whatsoever.
2.) My iphone will literally randomly access different apps screens and files on my phone--and even dial numbers from my contact list all without me touching it. I can literally watch as certain buttons are highlighted and it will randomly sweep through different apps, numbers, emails, etc. When I visited my local iphone store and told them what it was doing I was told it is literally impossible for that problem to occur. I left with the same phone and the same problem....This is not happening while the phone is buried in the bottom of my gym bag or in my pocket and there is a possibility that the screen is accidentally being pushed. This is happening while the phone is sitting on my desk, kitchen counter, etc. completely unfettered. I have had it with Apple and with the iphone and am at my wits end.