Tyre rubbing against frame Bicycle : Fixie, it came with fixed gear and freewheel, I use the freewheel cog.

Today, after a tire and tube replacement it now rubs on the frame. The new tire is a little bit bigger but this problem existed even with the old smaller tire, just that I managed to keep it clear. It rubs onto either one side, no matter what I do. (Refer to pictures 3, 3a and 4)

Also, I have been told that the frame is bent, which I actually don't buy since I have never got into an accident, unless it's a factory defect. (Refer to pictures 1 and 1a)

My brother decided to tinker around with the screws between the frame and rim housing, they now both spin whenever one side is being screwed, what do these screws do anyway? (Refer to picture 2 and 2a)

What are your suggestions or options that I have?
What you call screws are really axle nuts.  Fortunately your hub appears to contain sealed cartridge bearings, so there are no cups or cones to tweak up.  The axle nuts should be done up close to the bearings and then locked in place against each other.

Your frame has horizontal dropouts so try sliding the axle back a bit to see if that provides enough space for the tyre between the chainstays.
I can't comment on the chain length, never owned or worked on a single-speed bike.  

Otherwise, the tyre/tire is too wide for the frame.  Its possible the rims are not original either and the current ones are a bit larger in circumference than the frame was built for.  Sell off your trendy red tyre and go for a rear tyre that is smaller in width by 5 mm.   Or store it away as a replacement for the front tyre.