Tyre rubbing against frame
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.