I'm no RandyT but hopefully I can help a bit

You are correct that you will need 57 outputs for the RGB's and you will need also need to power the coin door clusters. Do you want to control the clusters too?
What are the voltage of the clusters? I'm guessing they will need 12v?
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If that is the case and you do want to control the clusters you will need two LED Wiz's and one LED Wiz+GP.
You could configure the first LED Wiz to run 5v and use the to control the first 10 RGB's using 30 of the outputs.
You could then configure the 2nd LED Wiz so that banks 1-8, 9-16 and 17-24 run at 5v and connect the remaining 8 button RGB's to run off that. You would then need to configure bank 24-32 to run at 12v and connect the two coin door clusters to two of the outputs (say 24 and 25).
You still have the trackball to light so you could run that from the LED-Wiz+GP using thee of the outputs.
This would leave you 2 free 5v outputs on the first LED Wiz, two 12v outputs on the second LED Wiz (from the 12v bank you created) and 13 5v outputs on the LED Wiz+GP.
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If you don't wish to control the coin door clusters you can just hook them up to a 12v molex (if they are 12v lights, you may need to use resistor(s) to bring the power down if they are 6v etc but I imagine they will be 12v)
In this scenario you will still need two LED Wiz's to control the 19 RGBs.
Connect the first ten button RGBs to LED Wiz one and the other eight button RGBs and trackball RGB to LED Wiz two.
This would leave you two 5v outputs on LED Wiz one and five 5v outputs on LED Wiz two.
You wouldn't need a LED Wiz+GP but you would still need to connect the trackball buttons. Its probably a bit over kill and spendy to get a encoder just for two buttons. I would suggest hacking the buttons from a mouse or a cheap gamepad, Randy stocks hackable USB gamepads for about $6.
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In scenario one, you wouldn't need to power LED Wiz one as it would take the 5v from USB, LED Wiz two would want 12v (Yellow and black ground) from a molex for bank 24-32. The LED Wiz+GP wouldn't need power as it too would take 5v from USB.
In scenario two, neither LED Wiz would need external power as it would take it from USB. You would just need to give the clusters 12v from a molex.
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As for wire, we are not talking huge voltages so you probably just want to use some stranded ~22AWG, the same you would use for wiring controls. Most home electronics stores (Maplins, Radioshack etc) should carry it.
Hope that helps a little.