Chekov lost a chess match to a Betazoid, he's Russian get it?
I never saw that scene as having anything to do with him being Russian. I'd attribute it more to Chekov usually being a comic relief character.
As I recall, the TUC novelization has an early scene where Kirk has to deal with the fact that Carol was seriously injured in a recent Klingon attack, and that provides some broader context for his discomfort at the command briefing. Is it possible this was at one point part of a working script, or mainly just used for the novel?
Yes, that was added by J.M. Dillard to further "justify" Kirk's hatred of the Klingons in TUC. Because I guess fighting Klingons for 30 years and having his son murdered by a Klingon wasn't enough.
In the Meyer/Flynn script of TUC, Kirk is in bed with Carol Marcus. She talks about how she's happy to have him home on Earth once & for all, and Kirk replies that with his retirement pay, he can barely afford to cross the street. Then a mysterious Federation envoy with a glowing hand comes to Kirk's door and summons him back to Starfleet. Kirk leaves over Carol's protests of "...But you're retired!"
Kirk then goes about rounding up his other crew members. McCoy is at a party full of doctors, half drunk, and McCoy tells off one of the other doctors when he overhears him getting indignant about a patient asking him to make a house call.
Scotty is giving a lecture at the Academy, where he's showing off the Klingon Bird of Prey that Kirk & co. brought home in STIV. He says something like there are still secrets of this ship that we haven't discovered, but since Klingons don't defect, we'll take what we can get.
Uhura, as said above, was hosting a call-in show with people from across the Federation calling in. When Kirk shows up to summon her, she says, "How much do I owe you?"
Kirk finds Chekov in a chess club, where he's bored and losing his current match. Kirk says, "Then I'm your fairy godmother," and points out that Chekov's opponent is a Betazoid, and therefore has been reading his mind for the entire match. I don't think it's explained how Kirk realized that Chekov's opponent was a Betazoid.
Sulu was a cab driver in a flying cab, which is very
WTF? 
, I agree. You'd think he'd be a test pilot or something. I guess they added the Captain of the
Excelsior thing in subsequent drafts.
Spock of course is revealed in the Starfleet briefing scene in the script. In Flynn's
The Fearful Summons novel, Kirk gets Spock on Vulcan, where Spock is playing Polonius in a Vulcan production of
Hamlet. I'm not positive, but I'm assuming that that scene was original to Flynn.
