IPhone 5 screen unresponsive?

I had my previous iPhone 5 for just under a year when I had to have it replaced 3 weeks ago due to screen faults. Everything worked perfectly until this morning when I tried to check my bus timetable - the entire phone froze and I had to restart it. Then I tried to send an email and the keyboard kept typing letters I hadn't pressed, and repeating the letters I had 10 times. I shut down the app I was using and tried to open another, only to find that the screen had become completely unresponsive. Once I had restarted it again, the same thing kept happening.
Does anybody have any idea what this could be or how I can fix it?
iPhone 5, iOS 7.0.6

Added (1). I have checked my storage availability and it is still at 7.9GB

Maybe you're running low on memory. Because your phone can become slower if you start to use a lot. If you wanna check how much memory you have left, go to Settings>General>About then look what's available. If it's less than 1GB I prefer you delete some apps, or some pictures and see if your phone works any faster.

Try this two methods:

Press and hold the home and power buttons until the Apple icon is displayed, then let go.

Method 2:
Reset, hold both home and power buttons for about 10 seconds until the iPhone restarts itself. Then when up and running, press home button twice quickly, then swipe up and away all the windows of running apps.