HI @jsanbower, thanks for reaching out. Great question and we appreciate the photos. I'll address your questions, below:
Nope, this is not a 2 wire system; it's an indexing valve as your research confirms.
All of the major irrigation companies offer quality pumps; I'd recommend comparing reviews online to make your decision. However, the model of the pump relay you need depends on the pump requirements. Do you happen to know the make or model of your pump?
Since the HydroTek is currently turning on your pump, you'll want to connect the blue and black wires to the pump relay and add wiring between the Iro to the pump relay. Make sure that the total draw of the Pump Start Relay does not exceed 1.3 amps at 24 VAC, 60 Hz. The controller does not provide main power for a pump.
This is more of a recommendation. While most of the time the distance between the controller and pump start relay doesn't create any problems, every once in a while the wires on the 24V coil can allow electromagnetic noise to travel back to the controller. This noise is made when the contacts on the pump start relay make and break the connection. As such, it's recommended to use a minimum distance of 15ft between the controller and the relay, to eliminate any electromagnetic noise (this applies to any/all high voltage devices).
Hope this helps. Feel free to reply with any additional questions.