A lot of questions about the games have been answered by the writer himself here.
How did the Rabbit end up in the same hole he dumped Alice in?
This troper was under the impression that he fell into it as well and somehow was knocked out. Otherwise, perhaps the hole is the standard place for dispensing agents and the rabbit was supposed to be stuck down there until he died.
He tracked her down to kill her, they fought, she won. The corpse was an unrelated person (possibly Mary Ann).
How was Lewis connected to Dr. Burr and the Oystercloud?
He was a scientist who aided in the creation of oystercloud. They killed him to surpress knowledge of oystercloud.
According to logs in the second game, he was arrested for stealing scientific samples, and ate the evidence when they arrested him. Judging by his corpse and what's inside it, it wasn't the best decision.
Was the Club a front for Hatter's killer circle? Since Dr. Burr's lab was located in its basement...
They used Hatter's killer circle, but were not equal to his killer circle, as shown by the fact that the hatter was imprisoned. It was a front for the queen's activities, and the Queen was funding Burr's research.
I was replaying Alice is Dead 2, and I was like "Why does The Hatter always look so angry?", then I moved closer towards the screen and I was like "Wait... is that hair? No, maybe they're his eyebrows, no, wait, it's hair, no.... it's not..." and I kept pondering about it, and I don't know if I'm seeing things. What do the tropers think about it?◊
Seeing as the Info Dump post no longer seems to exist...
What was oystercloud supposed to do?
Who was the corpse, seeing as it wasn't Alice?
Oh dear. That's an awful lot of lost knowledge. To put it in very simple terms, since I don't feel like writing a wall of text right now: Project Oystercloud was some sort of teleportation/dimensional movement experiment orchestrated by the Red Queen for some purpose. It apparently moved things between Wonderland and some "High Ground" or "High Place" or whatever it was called. If Wonderland depicted in the game is not our own world seen through the Rabbit's lens of mental illness (which is a realistic possibility), then Highground may very well be this world. Some of the subjects for the Oystercloud experiments were cats, but they always came back dead and horrifically mutated/mutilated. Then Doctor Burr's daughter accidentally touched something she shouldn't have just at the wrong time, and the Cheshire Cat was born, a sort of ghostly entity that appears to exist in some strange quantum flux, not completely real and physical yet not anything else, half cat, half girl, half eldritch abomination.
As for the infodump, though it's been deleted, you can find the archived version here. The author explains almost all of the headscratchers related to this game.
The corpse was just some random woman dressed up like Alice.
Uhm, maybe I'm just a bad gamer but where the devil is the Cheshire Cat mentioned? I've tried all the games, finished the third one three times but I haven't met her at all, only saw Burr's corpse with the explanatiosn about the accident.
Look through the glowing hole in the wall in the second game, and it will appear. It's not identified explicitly as the Cat (at least, not that I could find).