The servo drives appear to be Servo Dynamics 1225 or 1560 (I think those are the correct part numbers, going from my poor memory here) My machine has the same drives and servo motors. I have the pinouts for the drives if you need them. TomKerekes is correct on the command signal wiring if those are SD drives.
All of the servo system and controls should be supplied from 2 legs of the 3 phase input, thus single phase. The servo supply transformer should be powered up by an OPTO22 solid state relay that is controlled by the control circuit. They used these OPTO22 SSRs because they have zero crossing turn on and thus mitigate the inrush current somewhat. On my machine, the entire control panel plugs into a 120V outlet on the wall. The wiring to the VFD/spindle motor is totally separate. I also just went through a BP retrofit that has a control panel that looks pretty much like yours.
The only things on that machine that should be 3 phase are the spindle and possibly the coolant pump. According to the documentation on the Automation Direct website the GS3 2010 requires 3 phase input. But I don't know if supplying it with single phase would throw an error. Some 3 phase input VFDs will work if you connect one single phase leg to L1, the other single phase leg to L2, and put a jumper between L2 and L3. Try this at your own risk, I have never done it.
I can't help with the Dynomotion stuff, but Tom is an expert on that. But maybe I can be of some help with the servo wiring.