Pike was the captain of the Enterprise in the original Star Trek Pilot, which was dumped and reshot.
Pike was captain, Spock was science officer, but Majel Barret was the First Officer. NBC junked that pilot and they shot a new one with Kirk as Captain, Spock as first officer (still also science officer I suppose), and Majel Barret became Nurse Chapel and the voice of the computer (and of course Gene Roddenberry’s wife).
Pike shows up in an episode of the show later, played by a different actor in heavy makeup (he’s been badly injured and is stuck in a high tech wheelchair.) Pike in the wheelchair at the end of the movie is a shout out to that, although movie Pike isn’t nearly as badly injured.
Was there ever a Spock/Uhura romance? Was Chekov always good at video games?
Nooo. Look, putting a black woman on a TV show at all during that period was kind of a big deal, especially one not playing a maid or something. Putting her into an interracial romance (interspecies, whatever)…nooo. There was one episode where…I can’t remember the plot, to be honest, but she kind of flirts with this black/african crew member on the ship, who I think is an illusion or a hallucination or something. And in another episode mind control aliens make Kirk and Uhura kiss (the kiss was cheated/faked with camera angles so that NBC could cover their asses if necessary), and that was a controversial big deal, but the producers just fell back on “They were being forced by aliens to do it against their will”. And in the evil Mirror Universe episode I think there’s some notion Evil Sulu is into Evil Uhura. And in the original series I think Nurse Chapel has a thing for Spock, but I’m not sure how much that was reciprocated.
To be honest I don’t really remember Chekov getting much development in the series. He was proud of being Russian, and he was proud and protective of Kirk and the Enterprise, but I don’t remember him having other notable qualities. Sulu being a fencer is a shout out to the original series, in an episode where a bunch of the crew get space drunk from some disease and run around acting wild, Sulu runs around shirtless with a sword, challenging people to duels and such. Oh, actually come to think of it, I think drunk Sulu grabs on Uhura in this episode too.
Was anything about the way this crew comes together the same as the original continuity other than who’s left flying the Enterprise by the end? Was that sort of thing ever really even addressed?
I can’t speak for comics and novels and whatnot, but I don’t think the show or the movies ever really address how the crew first comes together, no.