The West Wing: A lot of people have complained about how Rob Lowe's feud with the show prevented him from appearing at Leo's funeral or the season 7 prologue. While I don't have an explanation for the last one, the first makes perfect sense: Sam is Deputy Chief of Staff now. If Josh, Santos, and (assuming she's still National Security Advisor) Kate are all at the opening of the Bartlett Library, someone has to stay and both run the West Wing, and stay on top of international issues. Sam's levelheaded (at least in comparison to characters like Josh), knows a lot about both the West Wing and international issues, and is in a high-enough position of authority to be able to act if anything comes up, and is thus the perfect person to leave behind and in charge. He wouldn't have been there even if Rob Lowe hadn't been being a pain. - Ambar Son of Deshar
With regard to Leo's funeral, he probably was there; we just didn't see him. There were a lot of people paying their respects in that church.
Another West Wing example: while this troper always has thought Bartlet's demolition of Ritchie during the season 4 debate was a Crowning Moment Of Awesome, she also felt that it needed a little Willing Suspension of Disbelief for Bartlet's incredibly smart-alecky remarks to be acceptable for political debate. Then later, putting together Bartlet's other public moments in earlier episodes, the episode "Let Bartlet Be Bartlet", the charge in "Manchester Part II" about raising the level of debate in the country, the summer-long gap between seasons 3 and 4; and Josh's remark that Bartlet was going to look arrogant no matter what, and they should just focus on making sure no one started demanding he act less arrogant; it hit her: status quo is not God: Bartlet, in the minds of regular people who don't work in the White House, must have been socially revolutionary; he must have been the very President who changed all the norms we, in the real-life universe, accept about political etiquette. After all, nothing Bartlet said would have been unacceptably snarky in, say, British politics, would it?
Another example is the way the election cycle is explicitly offset by two years (Bartlet was elected in 1998). At first this looks like a Hand Wave, but when you realize that the last real-life President ever mentioned in the show was Nixon, the simple explanation is that in the West Wing universe, something other than the events of real life must have happened upon Nixon's resignation in 1974 (such as a special election or an overhaul of the election process), causing all subsequent elections to be two years off the previously-established pattern. This would also explain a lot about the highly idealistic and direct form of politics practiced in-series.
For a lot of people (this troper included) it seemed a little forced and unrealistic when CJ became Chief of Staff on Leo's recommendation after Leo had to step down. However, Leo was the person who recognized CJ's talent in the first place and insisted she be a part of the Senior Staff. When told about this CJ was very flattered that he wanted her. It seems that Leo had her eye on her all along, got her a position that used her skills and gave her an opportunity to understand the workings of the Whitehouse. After realizing that it suddenly seems a lot less forced, given that Leo handpicked CJ and that CJ was so keen to take the job of Press Secretary.
There's more to it than that, though. Her role as Press Secretary meant that she had to be active with people both inside (the reporters in the White House) and outside (pretty much anything public), she had to be across all issues at all times, and she had to manage any crisis that came up. Meanwhile, although Josh was Deputy Chief of Staff, his primary job was as a congressional liaison, a role he was particularly good at - had he been promoted to Chief of Staff, he would have been unable to continue in the role he was most skilled in, and there would have been no obvious mechanism for his replacement. Similarly, Toby's (and Will's) field of expertise was speechwriting, and as mentioned earlier in the series, there are few who can actually do the job. In short, CJ was not just handpicked by Leo to begin with, she was also most qualified and the only reasonable candidate within the West Wing. And because of the timing, they couldn't possibly have found a new Chief from outside of the West Wing quickly enough to get them up to speed at that time.
Josh was far more qualified than CJ and it's unlikely that Leo would have chosen him as his deputy if he didn't think Josh had the potential to be a good Chief of Staff. However, if Josh served as Chief of Staff under Bartlet he would only hold the job for the relatively short timespan left of that administration. The way I saw it was that Leo wanted Josh to go out and find Bartlet's replacement and then serve as his Chief of Staff for however long he could. I think he saw Josh as a younger version of himself and, as Bartlet said, the future. Josh' reaction to CJ's promotion and what he does in the following episodes seems to support that theory to me and suggest that Josh was well aware of Leo's intentions.
"The Midterms". After the events of "In the Shadow of Two Gunmen", anyone who had Josh's gunshot wounds would require months of recovery. Not wanting to lose a core character for most of the season, what's a writer to do? Make a single episode span three months, so everyone's back at work by the next episode.
Allison Janney was most likely absent from "17 People" due to contractual reasons, but in universe the reason CJ is not there is because she is still in Napa celebrating her dad's birthday.
When Secret Service agent Simon Donovan is gunned down in a convenience store in the third season finale, the shopkeeper knocks over a bouquet of red and white roses. At the same time, the rest of the cast is attending a performance of a play based on the Wars of the Roses, a civil war between the House of Lancaster (whose symbol was a red rose) and the House of York (whose symbol was a white rose) for the English throne.