Hi Julian,
In regards to the replacement screen, is everything assembled correctly? Did you buy a display assembly or did you just get the LCD in which case you must transfer the front facing camera and the metal back piece of the digitizer?
Did you keep track of which screws go where when replacing the aluminum cover that goes over the display cables? If you mix those up there is a possibility for a screw to go through the logic board and cause issues.
Before removing more components I would check the following:
The replacement LCD is fully assembled, check the ribbon cables for wear or tears.
Check the screws to make sure they have been ordered correctly.
Once the screen is replaced, make sure to do a full restart (Lock button and home button held down).