The West Wing/WMG

Josiah Bartlet is Dolores Landingham's padawan
The episode "Two Cathedrals" shows a mentor-student relationship between the two, with the older woman latching onto a young man in whom she sees incredible potential, becoming more of a parental figure to him than his father, insisting upon his calling her by a formal title, and grooming him to play a major role in reshaping the world into what, from her point of view at least, is a better place. The clincher is at the end of the episode, when Jed shouts for her after her death and she shows up to give him advice much like Obi-Wan did for Luke after his death. Jed doesn't seem at all surprised to see her; and their conversation acknowledges her death, so it's taking place after she's passed on. Not once does the President question his sanity because of this, so it's not out of the question that he was expecting it. Force ghost, anyone?
 * In this, there's a tenuous possibility that Jed is short not for Josiah, but for Jedi.
 * This raises the question of who Jed's padawan is. Perhaps it's Toby Ziegler. A brooding, ill-tempered man who has committed a serious crime for what looked like excellent reasons at the time, betraying Jed in the process. He has also fathered twins, a boy and a girl. This doesn't look good.
 * It's Charlie Young, with Jed's handing over his father's copy of the constitution in the last episode marking his graduation. The gift of Paul Revere's knives in the first season marked the start of his training.
 * Sam fits the bill too. After all, Jed did express a desire to see him become president one day. Josh is clearly Leo's padawan, but we don't know who trained Leo.

Will Bailey and Elsie Snuffin had an incestuous relationship
When Sam meets Will and Elsie working on their Californian campaign, he instantly assumes that they are, or were, romantically involved. It's a reasonable assumption for an Aaron Sorkin show - two young, attractive people of the appropriate gender working together are almost guaranteed to have some kind of romantic history, the more excruciating the better. But Will describes Elsie as his "stepsister", so that's out. And yet - why make her a stepsister? For the same reason Boone and Shannon on Lost were stepsiblings - so they could have a shocking incestuous affair without genetic issues. This would explain why Will is so reluctant to work in the White House - he's afraid of being found out and causing yet another scandal for the administration. Presumably, this storyline was quietly dropped when Aaron Sorkin left the show, and Elsie with him.

Josiah Bartlet secretly agreed with Toby's shuttle leak
The president has broad powers when it comes to federal crimes - he can completely pardon a criminal or reduce his sentence to any degree. It would be possible for Bartlet to let Toby's conviction stand but reduce his punishment to a simple fine, as in the real-life Scooter Libby case. If Bartlet genuinely thought Toby was wrong but wanted to spare him a prison sentence, that would be the logical thing to do. But a freeze-frame of Bartlet's pardon shows that Toby's conviction is completely overturned. Therefore, Bartlet agreed that the shuttle should have been made public. All his protestations to the contrary were political posturing. Even when he rages at Toby in private, he's just ensuring that Toby can honestly say that the president was angry at him for the leak.
 * Plus, it's not in complete private. White House Counsel Oliver Babish insisted on his presence, even though Toby and Jed both expressed their preference he not be there.
 * As for the private raging, it seems totally consistent with Bartlet's character--and his and Toby's sometimes fraught relationship--that he would be that genuinely pissed about the betrayal even while agreeing with the leak ideologically. The harangue was more personal than political in orientation.
 * Agreed. Saving a group of astronauts' lives at the price of revealing the existence of a top secret shuttle is exactly the kind of thing Bartlet would do if it were just him making the decision. But since he's the President, that's not the kind of choice he can make. And the fact that someone on his senior staff, someone he's known and trusted for years, deliberately went behind his back and took that choice away from him probably ranks as the most damning sort of betrayal. But even then, he doesn't fire Toby because of personal reasons; indeed, when he passes down his judgment, he's much calmer than in the rest of the conversation, telling Toby, "I can't accept your resignation for this, Toby. I have to fire you." It shows that he's not firing Toby because of his betrayal of Bartlet personally, but because the magnitude of his security breech was so great that the public wouldn't accept anything less as a punishment. Letting Toby resign would show tact approval for his actions in the public eye, which would be political suicide. And even the decision to write a pardon came at (almost literally) the last minute.

Toby's ex-wife is Jewish
Sure, her surname is "Wyatt" and she's a redhead. But we know that her son had a bris ceremony, which means she and Toby consider him Jewish. Since it's traditionally the religion of the mother that determines whether a child is Jewish, Andrea Wyatt must be Jewish herself.
 * She could have converted to Judaism when they first married.
 * More liberal denominations of Judaism don't generally care which parent is Jewish.

The Entire West Wing Universe is an Alternate Reality...
Quasi-canonical because the show features skewed election cycles to the ones in the real world, but over the years many fans have suggested various jumping off points for when the timeline skew happened. A popular one is that Nixon was impeached and a special election was held, restarting the 4-year Presidential cycle at 1974. Perhaps, since Ford didn't pardon Nixon, he wins the 1976 election instead of losing narrowly as in our universe. By the time Ronald Reagan gets a chance to be elected, he's considered far too old or showing clear signs of Alzheimner's. There's never a "Reagan Revolution," and the whole US political climate is less viciously partisan. This explains why everyone seems much more polite and reasonable on The West Wing, and why a Democratic White House could hire two Republican lawyers, which would be impossible (or at least controversial) in our universe.

This, by the way, also explains why they inexplicably have a super secret military space shuttle in the 6th season. The space program had more support and more competent suppliers in this timeline.
 * But then how come they only have two civilian shuttles as opposed to three? CJ is told that "ONE's missing half its wing assembly and the other ONE's undergoing a complete tile replacement."
 * Ripples in the timeline; presumably a different president cut back the shuttle program so that there were only two made rather than three. Alternatively, considering they have at least one military shuttle in the West Wing universe, presumably resources were split off into that project instead of a third civilian shuttle.
 * The most recent public official from our reality who's mentioned on the show is Donald Rumsfeld, as Chief of Staff to Gerald Ford. That places limits on when the timelines diverged.
 * One theory is that Nixon resigned as in our timeline and Gerald Ford decided that he was not a legitimate successor since he had not been elected Vice-President or President. Through some deft legal and political maneuvering, he was able to order a special election for President in November 1974.
 * This is the generally accepted explanation. Ford then won the 1974 election, then lost in 1978 to D W Newman (a Jimmy Carter analogue), who would in turn lose in 1982 to Owen Lassiter (a Ronald Reagan analogue), who would serve two terms before being succeeded by an unnamed centrist Republican (judging by how Leo was a cabinet member despite being a Democrat), who also served two terms before Bartlet's first win in 1998.

If the divergence point is just a few years earlier, then the 25th Amendment goof could also be resolved.
 * A subplot running throughout season 2 is: When Bartlett underwent anesthesia after being shot, he did not sign a letter to allow Hoynes to assume the presidential powers. Without that letter, there was no-one in command.
 * The Goof: Section 4 of the 25th Amendment specifically allows the VP and the cabinet to assume power in this kind of situation.
 * The Solution: The timeline diverged in 1965. The only difference prior to the Ford presidency is that, in the West Wing Universe, the 25th Amendment only had the first three sections.

The show's universe has its own The West Wing, and it depicts our universe
This was popular a while back in one alternate-history newsgroup; the only permitted discussion about recent events in our own timeline was in the form of discussions purportedly 'from' that universe, about the show.
 * It started out as an 'edgy' Dramedy for the first two seasons with the professionally effective, witty and charming "President" Clinton, whose one flaw was that he couldn't keep it in his pants if his life depended on it. A Retool for the 2001 season replaces him with slapstick "President" Bush (Bob Denver), who starts out as a slightly dim Ivy League legacy who benefited greatly from nepotism but is eventually Flanderized until everything he does is a disaster and he literally can't find his own hind part with the aid of military-grade GPS, let alone a map and compass. Amid declining ratings, the show was canceled after Denver's death in 2005. Rumors of a revival of the earlier, more serious version starring Will Smith remain just rumors now are going into their third season.
 * I stopped watching after episodes which dealt with the 2004 elections. The big issue that season had been how the war in the Middle East was going badly and then they expect us to believe that President goofball beat the war hero dude (can't remember the guy's name; Roy Scheider played him but was pretty wooden) it was clear the show had Jumped the Shark. I honestly thought he was going to be a Midseason Replacement.

Mandy's PTSD after the shootout at the end of the first season manifested as an ability to break the fourth wall.

 * She realized that she was present on the show in order to drive plots by making catastrophically bad decisions and then feeling bad about them, so she voluntarily put herself on a bus in search of a better life.

Jean Paul doesn't belong on the Karma Houdini page at all.
He hasn't been seen since slipping Zoe a roofie because he was taken to a CIA black site and remained there for the rest of the series.

Mandy was an Auror
She was assigned to protect the muggle President from a particularly dangerous Dark Wizard, ala Kingsley Shacklebolt in Half Blood Prince. When the wizard was defeated, she disappeared off the face of the earth.

Bruno is the father of Margaret's child
In The Black Vera Wang the two exchange a moment when Bruno gives her a fairly expensive looking necklace (the box even looks like the sort you would put an engagement ring in) and the two smile at each other.

Sam's unnamed fiancée in the final episode is Ainsley Hayes and the two will go on to be The West Wing universe’s version of Carville and Matalin.
There was romantic tension aplenty between the two when they first worked together, but neither wanted to sully their reputation by acting on it. Once they were both working in the private sector, they were free to date. Since they will both resume working in the White House under Santos, they are keeping it quiet until the actual wedding date is set, which is why they didn't appear together in the final episode. Sam was gearing up to confess to Josh, who was too busy to hear it at the time. Part of Sam’s motivation in encouraging Josh and Donna to finally get together was so that he and Ainsley wouldn’t be the only couple working together in the White House.

Mandy was swallowed by a crack in time.
In the fifth series of Doctor Who, a recurring plot element are 'cracks in time' which have fractured every single moment of reality and, if some poor unfortunate gets too close to one, ends up not just killing them, but erasing them from history altogether so that no one remembers they even existed. This would explain why a fairly significant member of the senior staff of the President of the United States can not only disappear, but no one ever mentions or alludes to her ever again; she's been wiped from reality.
 * Sam Seaborn and Ainsley Hayes suffered the same fate, but made it back when the Doctor rebooted the universe, hence everyone suddenly remembering them in season 7.

Jed Bartlett generates inspiring music
Or possibly Martin Sheen. It's the only logical explanation for how so many of the episodes end.

Mandy willingly quit her job at the White House to...

 * ...return to her hometown in North Carolina with her teenaged son. Her real name was Karen, and "Mandy" was just a professional pseudonym she took up while working in Washington. Bartlett and the White House security had to know "Mandy"/Karen's real name, of course, for security purposes (because after all, this is The White House), but the public only knew her as "Mandy" in order to protect her privacy and allow her son to have a relatively normal life. Eventually when Karen moved back to Tree Hill, nobody seemed to even realize that she had been one of Bartlett's staff members.

Josh Lyman is only kept on staff in fear of blackmail
The man has done so much collateral damage to their administration, the only reason he is kept on is because if he was let go, he would expose every little dirty secret and flaw in the Bartlet administration. It would be like the business with the Vice President, but worse, as he is more connected to all of the staff, and knows more of their dirty laundry. The only reason he's tolerated is because his goals are the same as theirs, and his support staff(Donna) actually gets things done.