The additional socket might also be for some custom font ROM or similar. I'm not 100% sure but I think HD61830 had this kind possibility.
But that controller board you have looks like it should be enough to drive the display. That 10 pin connector found from both the controller board and display is most likely meant to be connected together. Just find some cable for it and also find out which way to connect the cable. If you trace the HD61830 pin names where the connector pins go and also trace some of the display connector pins going to the IC's in the display we should be able to tell you how to connect it.
Also that other connector should have signals for connecting to LPT port and voltages. Again if you trace the traces going to the HD61830 we should be able to tell you how to connect things.
I know I could see most from your pictures already but I'll save my time and let you trace where things go and do all the work and give us easier presentation of it to work with.