When it comes to the wiring error, it seems unlikley since the backpack pins are connected to the right spots into the display from 1 to 16.
Back in reply #5, just before we were sidetracked, LarryD recommended that you post a picture. That remains a good idea because it will help us determine if you really did everything as you intended. For example, you are connecting an 18-pin device to a 16-pin backpack. This should work as long as you really did implement pins 1 <--> 16 as intended and not 3 <--> 18 and this is a place where another set of eyes could help.
I might try to desolder the backpack from the display and see if it works with normal wires.
I was going to suggest this but it is a lot of work if you don't have decent de-soldering equipment. It would be a good idea for you to check your soldering for cold joints, bridges, etc.
If if the display had power, the backlight would work, it would show black squares instead of characters, ...
Not necessarily. The backlight is turned on by software and since it is off the black squares may be difficult or impossible to see (I think more people will agree with me about this, including liudr, than will disagree).
... and the green and blue backlight on pin 17 and 18 would work, since they're grounded trough the backpack.
I don't see how this could be the case since, the way I am picturing things, pins 17 and 18 are not connected to the backpack. Also, they would have to be powered, not grounded, since they are connected to the LED anodes.
Don