why did Kirk's crew and Kruge's crew need to activate the transporters at the same time for Kruge's boarding party?
This did not happen. Only Kirk's crew did any activating as far as we can tell. And supposedly this was because Kruge's own transporters were out of action (Kirk's ship did hurt Kruge's!), as we never saw any of the telltale Klingon red glitter in the process.
It confuses matters somewhat that Kruge's boarding party assembles in Kruge's transporter room. But where else should it assemble, on that cramped little ship? On the bridge, where they initially don their gear? That would mean allowing Kirk's transporters to probe into Kruge's most secure facility.
Conveniently, no transporter room set was built for the Klingon ship yet, not for this movie; Kruge's troops were not shown standing on Kruge's platform,
or next to it, so we can choose which version to believe in.
In any case, Kruge is the passive player there, waiting for Kirk's signal, and then just telling his team to expect a transport when Kirk sends said signal. It's not a pad-to-pad transport except for the incidental possibility of there being a pad even at the passive end...
What's the deal with Ambassador T'Pel's transport to the Devoras?
The E-D beamed the fake Ambassador to the Romulan ship using standard procedure, expecting no activity from the Romulan end. The process worked without a hitch, and the agent materialized on the Romulan ship. Simultaneously, the Romulans beamed back fake blood.
Our heroes didn't recognize this at first, and even Data's investigation went the complicated route of assuming that the Romulans
also grabbed the fake Ambassador with their own beam - even though nothing suggested anything had really gone wrong with O'Brien's own effort of beaming the fake Ambassador.
Examples of incidents where there really
is some receiving pad activity even when the transport is initiated by the sending pad are fairly uncommon. ST:TMP is a jumbled mess in more ways than one. But there's also the aforementioned "Dramatis Personae" where a Cardassian transporter finishes the job started by a Klingon machine that blows up in the middle of the process. A red glitter reassembly fails; an amber glitter reassembly subsequently succeeds.
Basically, it would seem that a transporter, any transporter, can turn a person into a blob of "phased matter", existing outside our traditional realm. It takes active effort to keep that person in this alternate, phased realm: fun with diagnostic cycles in "Relics", something very similar in "Counterpoint", etc.
(The "Dramatis Personae" example also rather suggests that "phased matter is phased matter is phased matter": a transporter doesn't create a transporter-specific package, but a
generic blob of phased matter, something that can subsequently be manipulated by any phased-matter-manipulating machine and not just those of suitable type or authorization.)
If there's no such activity, the person decays back to our realm - but this process may go wrong in several ways, and if that happens, an active transporter can actually midwife the person back in a controlled fashion. In essence, transporters are machines that can access the phased realm and manipulate things that exist there (even in fairly complex ways) - but for a normal transport, they don't
have to do so. They just have to push the person into phased space and send him in the desired direction.
For the transporter chain idea, we could postulate that one way to manipulate the phased package is to give it some extra push in the desired direction, while another is to prolong its phasing. Booster stations could then exist in chains that never rematerialize or otherwise manipulate the transportee - they just push him forward in phased space
at an ever-accelerating "speed", possibly defeating the speed and distance limitations of single-transporter operations.
What doesn't fit this picture? Well, possibly ST:TMP. But that's an infamous outlier in any case. And we could argue that what happens there is that Starfleet sends, the
Enterprise is passive, and what actually goes wrong is that a power spike
unrelated to transporter operations results from the small explosion at Main Engineering and messes up the incoming signal. At that point, Rand has to attempt active recovery of the poor sods in phased space, and this attempt notoriously fails, despite Starfleet "boosting their signal" and then trying to pull back what they sent. It's at first golf with a ball of phased matter, with the
Enterprise as the hole; then turns into tennis or volleyball when the
Enterprise isn't ready to receive; but soon decays into football grabbing which ends poorly for the pigskin.
Timo Saloniemi