No no no no....
Do not remove the Header pins on the display, when you do, you are only making it harder to yourself to fix it.
True, the pins on the display are not long enough to make a decent connetion with a ide-connector. I had this problem too.
This is what I did.
I cutted the connector so that it would only leave me a 9x2 connector. Then,
Just 'shave' off a bit of the ide-connector it's bottom (2-3mm should be fine, when you go further than this you are able to see the pins IN the connector, when you see them, they always will make a decent connection) then it would slide on perfectly. put a ribbon cable on it, and you are ready to go. I tested this with a multimeter if all the pins had a decent connection, and they had one so it worked.
You are right about the pot-meter, it's a bit strange to connect it like this, but I connected it like the diagram (btw. yep that's the correct schematic), and It worked. You should be able to put the backlight at full brightness (which is really bright) or almost completely off.
Hope this helps you out !!!