Lassard knows full well that Harris hates him...
...and he takes great pleasure in making the Jerkass's life as miserable as possible. That's why he's always so happy to see him. Tackleberry was discharged from the military under Section 8.
When the police academy drops its standards, Tackleberry is taken on without question. Zed was allowed to join the police academy because he never killed or seriously injured a person...
As the crazy punk he was, he was more interested in just scaring them and doing whatever he liked. Despite his violent behavior he never committed harsher crimes and during his knife fight with Mahoney, he only toyed with him. He also pleaded guilty in court and was released early for good behavior. The reason he shows up at the academy in 3 is someone gave him the same chance as Mahoney to prove himself as willing to change.- Maybe that someone was Captain Reed from the first film.
Harris and Mouser worked together before.
Due to their similar personalities (and their mutual dislike of Mahoney and the misfits). Mauser may have been the one who assigned Proctor to Harris sometime between 3 and 4. And Harris had Blankes and Copeland work for Mauser in 3. Mahoney eventually becomes Commandant of the Academy
His being Put on a Bus was, in fact, him being transferred to other precincts in order to gain experience in senior ranks in order to become Commandant. The Blue Oyster is a franchise
The location of the bar changes between films. It's unlikely that they've been chased out since their clientele is implied to be gay. So, who's to say there's only one Blue Oyster? Why not a chain of them in the vein of a gay Hooter's? It would also explain why Proctor knows exactly where it is in the second film yet stumbles blindly into it in the third and is tricked into going there in the fourth. Proctor got demoted for running around naked in the third film
He's shown with Captain bars in Back In Training, yet is a Lieutenant in the next film. Someone recognised him and reported him to Commissioner Hurst. He outright states that he is related to Harris through marriage, so he's only demoted to Lieutenant instead of something more severe. Zed and Sweetchuck disappear after the fourth movie because one of them killed the other
Zed and Sweetchuck don't appear in the series after Sweetchuck attacks Zed in the air show. Maybe Sweetchuck renewed his attack five minutes after credits and one of them ends up killing the other and the remaining one ends up in jail. That would explain their absence and the lack of anyone talking about them after that.