NASA report SP-6, chapter 2 describes the tracking network in detail. It provides this overview schematic explaining how the various stations communicated with each other:
(source: NASA SP-6, chapter 2, figure 2-8)
Communications were via the following channels:
The submarine lines and land lines were direct-line links, ...
I can't provide a definitive answer, but I think the main limiting factor, beyond mere physical matters like getting the spacecraft there or having it be able to transmit data at a useful rate for accomplishing the same thing as on the Moon, is indeed the speed of light itself.
Anything that requires manual control of the spacecraft (such as keeping them ...
It's mostly a terminology question and an application question. As you've correctly pointed out, "delay-Doppler" is the term used for radar tracking of uncooperative systems. For cooperative spacecraft tracking, the term "Doppler" measurement or "range rate" is used. The underlying physical principals are the same.