Just to double check, did you check the Wiki link I gave you? It outlines the pins in order.
Neverthless, if a Gorf joystick that is "properly wired" can be connected to the "KeyWhiz" or "IPAC" then I can't see why it would not be the same for a "Hydrogen" controller. After all, don't all of them send +5V to the controls?
Short answer is, yes.
Long answer is;
Not sure why you're missing the first three pins. The pins outlined in blue in the attached photo are pins, in order, 1, 3, 4, 5 (there is no pin 2) of J1. The pins outlined in red are the three pins you seem to be missing. Yellow is what you have. With the Red/Yellow outline you'll count down 1-3, no 4, 5-10. Flip the board over and note which pins have their corresponding solder points.
Yes, the controller boards do have +5 on their control pins, but this is usually a signal line, not a supply line. You'll need to supply clean +5v from somewhere like the USB bus or the appropriate pin from a power supply. Even a simple 5v wall wart for a cell phone works well (I add a regulator when I'm testing regardless since the GORF sticks are so old).
Neverthless, if a Gorf joystick that is "properly wired" can be connected to the "KeyWhiz" or "IPAC" then I can't see why it would not be the same for a "Hydrogen" controller. After all, don't all of them send +5V to the controls?
The short answer is that most controllers like the KeyWiz probably isn't going to have enough amperage from those pins to drive something like the GORF stick. That isn't how it's meant to be used anyways and I can't think of any way to power
and detect the GORF stick directly from a KeyWiz (The LED-Wiz+GP might work though) control line.
Supplying +5v to most of those pins isn't going to do what you want anyways. The way it works is that the GORF directional pins actually put out a +5v signal. When you pull the stick in a particular direction, the voltage is pulled down to or near 0v. Conveniently enough, this is exactly what controllers like the KeyWiz look for. It too puts out +5v signal and when this signal is pulled to 0v (or grounded via a switch), then it registered as a button press. The two seem to play nicely with each other. So you need to supply +5/ground to pins 10/7 and the GORF stick will be functional.
I don't know anything about the "Hydrogen" controller or how it works so I won't dare say it works with the GORF stick.