Going Postal (Discworld)/WMG

Vetinari paid for the Grand Trunk's repairs with the bounty on Reacher Gilt.
The Patrician is an Assassins' Guild graduate, he was at least as responsible for Gilt's death as he was for Lord Winder's, and it'd appeal to his sense of symmetry to collect on the Guild contract for that sole purpose.

Adora Belle is Anoia, the goddess of things stuck in drawers (or possibly could be an avatar or possessed by her).
Think about it—Moist has become obsessed with Anoia since the whole praying-for-money incident, said incident increased her worship dramatically, and when Anoia herself shows up in Wintersmith it's exactly the same character as Adora Belle.
 * Moist also points out that you're supposed to smell cigarette smoke in the presence of Anoia, and anywhere Adora Belle has been for more than ten minutes reeks of it.
 * In Going Postal, Mr Pump says this about Adora Bell: "Anghammarad Said She Reminded Him of Lela The Volcano Goddess, Who Smokes All The Time Because The God Of Rain Has Rained On Her Lava." And in Wintersmith, Anoia says "I used to be a volcano goddess [...] and the god of storms was always raining on my lava."
 * Also note: God of Rain. Moist Von Lipwig. Rains on Anoia/Lela/Adora Belle's lava.
 * Alternatively, Anoia appears in her current form because Moist had Spike in mind when he popularized her worship.
 * As did all her new followers, when they saw Anoia's favored postmaster propose to Adora Belle in public.
 * Also, Adora Bell is obsessed with freeing trapped tools AND she's the one who finds the key in the hidden draw in Lavish's desk, opens the secret cupboard, AND she's the one who proposes that they leave the contents on the floor. I think this qualifies as yet more Fridge Brilliance on Pratchett's part.
 * If you look at it metaphorically, Reacher Gilt and the other owners of the Grand Trunk were Stuck in the Drawer (i.e., the Trunk) preventing smooth business and efficiency. Adora Belle helps Moist unstick it by removing the offending object.
 * Plus, Spike should probably be dead or having major lung damage from the amount of cigarettes she smokes. Being a goddess would prevent that.
 * Moist got her attention with his Last Words. "I commend my soul to any god who can find it."

Vetinari is grooming Moist von Lipwig to be his successor.
If Vetinari wants something, he takes one thing to mold it into what he wants. He legalized the Thieves' Guild to regulate crime, then he took Vimes and made him forge a powerful new Watch. For now, the city is not only stable, but growing. In order for it to thrive he needs to revitalize the major public services. So he takes Moist and uses him to resurrect the Post Office, which becomes big business in and of itself. He then targets the banks and the tax office that we know of, planning to use Moist to do so. He would have used Reacher too, if he'd have cooperated. He'd have played the two off of each other to create one ideal successor.
 * As for why he'd want someone like Reacher or Moist, he knew stability was assured with his system in place, especially while Vimes or Carrot was around to keep the Watch together, so he saw that the city could need help in order to grow as well as it could. Thus he needed someone savvy enough to keep his system, know how to handle people and know how to get things done. Instead of Vetinari's subtlety he relied on a show biz personality to make the people go along with these ideas, leaving the old fashioned Guild leaders powerless to resist the change.

Vetinari is grooming both Moist and Sam Vimes to be his successor(s).
In more recent books, Vimes has started to show a streak of Magnificent Bastardry, which he even remarks on in Thud!. Knowing that Vetinari takes a personal interest in his development, it seems quite plausible that this was his intended goal all along. Vimes and Moist represent different elements of what makes his style of government successful. Vimes is focused, has a clear sense of right and wrong, is absolutely incorruptible and has a talent at upsetting established power players, but can be a bit hard-headed. Moist is a born con-man, good at manipulating people and getting them to like him and thinks outside the box all the time, but he's flighty and unreliable unless forced to focus on one task. It's possible that Vetinari either intends to make them compete indirectly to show which one would make a better successor or split up his power between them eventually. Also note how, in his interactions with each of them, he seems to channel the spirit of the other. With Vimes, he's gratuitously obfuscating and manipulative, almost trickster-like, while with Moist, he's a stern authority figure.
 * The main problem with this is that based on the events in Night Watch Sam is pretty similar in age to Vetinari (he was starting as a cop around the time Vetinari was starting as an assassin) so he's a bit to old to be a good successor.
 * Age may not be relative. The Patrician has had how many death attempts? He is seemingly more unlike himself/unhinged as the series goes on. Sam may only be the Plan B should the Patrician needs a replacement before Moist is ready.

If/When Vetinari cops it, Lipwig will be forced into the Patricianship via an assassination contract
Even if Vetinari is grooming Moist into the next Patrician, the chances are Moist isn't going to particuarly want the job, because it completely destroys any chance of reverting back to his old life (for one thing, they'd have to put his face on the stamps), and like the other positions Moist has taken, it's not exactly renowned for it's holder's longevity (unless of course you're Vetinari). Vetinari has probably set up a contract stating that he's to be assassinated if he refuses the Patricianship, and Moist will only be informed of this after Vetinari's death. Also has the bonus of stopping the other Guild Heads putting their own person up for the job, because by then it will so obviously have been Vetinari's plan, and nobody wants to know what happens when a plan of Vetinari's doesn't go his way, even if he is dead.