Hi Scott,
I have built several TDOA's over the years, this is actually a Time Difference Of Arrival (TDOA) circuit not Doppler. It rapidly switches between two antennas to measure time difference in the arrival of the signal between the two antennas to determine if it is to the left or right. Doppler electronically spins a circular array of antennas (creating the doppler effect) to determine a discrete direction pointing to a signal on a circular display that looks somewhat like a compass. Rather moot point in this case I suppose but that is the difference. I did not build mine exactly the way shown but rather used a PVC frame with BNC connectors for the antennas so I could try different combinations. Anyway... to your questions...
1.) The antenna connection to the radio in my case was a length of RG-174 soldered to the circuit board to a BNC connector which matches the radio I used for this project.
2.) The connection you are mentioning is a roughly 1" diameter coil of thin wire (I used wire wrap wire) with about 30 turns that straps to the radio's speaker area. This inductively couples the audio from the radio into the circuit. In this case the TDOA switches between the two dipole antennas at an audio rate, you hear a tone on the radio, that tone is used to determine which antenna is receiving the signal ahead of the other one (the phase of the audio tone changes as you turn the antennas). The tone nulls out when the antennas are pointed to the signal of interest. This circuit evaluates that tone to give you a left/right indication. Note on that coil, if the left/right indication is backwards, turn the coil the other way around. This seems to vary between radios.
3.) In my case I did not use a wood frame and wire for the dipole, not that that does not work just fine. I used a PVC frame I had built for a previous TDOA, it has BNC connectors on both ends so that I can try different antennas and uses four removable antennas to create the two dipoles. As far as length, roughly 1/4 wave for each leg of the dipoles (so they are half wave dipoles) at the frequency of interest (in this case the two meter band) is what you want to shoot for. The spacing between the two dipoles should also be roughly 1/4 wave. Note the "feedlines" lines from the PC board to the two dipoles need to be the same length, otherwise the directional indication will be off and you will not have an accurate indication of when you are pointed directly at the transmitter.
Hope that helps some, again, I kind of departed from the way the prototype was built. I am attaching a photo of mine.
Ron
WB5DYG