I accessed the remaining pins using an Arduino Proto Shield between the Arduino and the display, but I see you do not want to do that.
The pins that connect to the Arduino are surface-mount soldered on the back side of the display PCB. They are staggered, so the pitch would be an easy-to-solder 0.2 in. They are not marked though so you would have to count them carefully. You could also solder jumper wires directly to the bottom of the Arduino. Those are thru-hole, so the pitch would be 0.1".
The Arduino and display are 21mm tall, not counting the feet on the Arduino. With the proto shield installed the height is 37mm tall.