Other mistake: How could Egon get down to the ground floor so quickly? I mean, a few moments earlier, he poked a guest in the shoulder. And, why would he go back down to the first floor in the first place? They said that nothing happened on any floor except the twelfth.
Suggested correction: Egon probably was able to track Slimer to the ballroom immediately after he slimed Venkman, Egon never says he was on the first floor, he probably knew the layout of the hotel and picked up Slimer's trail heading for the ballroom.
But Egon says, "get down here right away, it just went into a ballroom." That line alone indicates he is already down there.
Egon can still say 'get down here right away" while he himself is still en route to the ballroom.
Somehow I don't believe a man as intelligent as Egon would use an improper term such as "get down here" unless he was already at that location. He more than likely would have said, "get down to" or "get down stairs." Speculation yes, but in terms of his character, I feel this makes sense.
Character mistake: Meeting Dana at the Lincoln Center fountain, Venkman tells her that Gozer was a deity worshipped in 6,000 BC by Hittites, Mesopotamians and Sumerians. There are no historical accounts of those specific populations dating back to that ancient of a date, or writing of any form for that matter. (00:43:40)
Suggested correction: Venkman said Mesopotamians, not Babylonians. The Mesopotamian civilization existed in 6500 BC so his information would be accurate in that aspect.
You are absolutely right, I have misquoted the movie there and I just fixed it thanks to your correction. "Mesopotamians" is just a generic denomination, though, so saying that Gozer would be worshipped by the Hittites in 6000 BC, and then list a generic name for the inhabitants of the area and then the Sumerians (generally accepted as the first organized civilization of the area, still much after the proposed date) doesn't seem to be quite accurate.
A valid point of view from both of you, however, it's established that Dr. Venkman is the least knowledgeable in paranormal history so it's possible he listed the groups out of order. As is rightly pointed out, Mesopotamia was a generic listing, like saying Americans, so maybe if he said Mesopotamians first, then the other groups, the quote would make sense?
I completely agree that he's the least knowledgeable of the bunch by far and it is well-established and reinforced in the same scene since he needs Dana's help to read "Hittites." Read, because he's not quoting from memory; he's reading (presumably something Egon or Ray had to write down for him). As you say, "if" he said Mesopotamians first, maybe it would have made sense (but it's not what he says), and even then, using your example, saying "in 1000 BC, Manitou was worshipped by the Sioux, the Americans and the Apache" just sounds wrong. (I know that by 'Americans' we mean generally the US population and feel free to add 'native' to it, then again Hittites established their civilization when the Sumerians were wiped out already, and that's 4 millennia past the date - it is what happens making a word salad in a small sentence that has to register just superficially for the audience).
Revealing mistake: You wouldn't want to sleep inside the Sedgewick hotel, and not because of its ongoing infestation at the 12th floor; a small food cart rolls into a wall and that alone makes it wobble. (00:33:20)
Other mistake: After the trio stormed off the library, they are walking around the campus. As Venkman giggles mocking the "Get her" lack-of-a-plan, to the right of the screen there's an elderly lady who keeps staring curiously right at the camera. (00:12:20)
Audio problem: When the Archbishop says "How are you, Lenny?" to the Mayor in his office, his lips don't match what he is saying. (01:15:55)
Continuity mistake: Distance between the guys when they find the ghost librarian decreases significantly between shots. (00:10:40)
Other mistake: As Gozer is flipping over the Ghostbusters, we can see some sort of ceiling decoration, but they are on top of a building and even the open portal door shows no roof above them.
Suggested correction: That's actually part of the building spire with clouds above it. Not a ceiling. The clouds are fake of course and not moving so it looks like a ceiling. You can see this spire again when he marshmallow man is fired upon by the ghostbusters and the explosion later.
Visible crew/equipment: When Ray pulls up in the Ectomobile (before it has been painted), Peter is walking over to the car and the camera is following him. In the reflection of the window, a piece of equipment can be seen (possibly a boom mic). (00:20:30)
Suggested correction: Possibly a street light. Judging by the shape of it.
Continuity mistake: When Winston says "I wanna make a phone call" to the guard in the jail, the guard closes the door of the cell twice. (01:10:45)
Suggested correction: In fairness, you do hear two noises, the second more subtle. I took it as the door taking an extra pull to properly close. You can argue that in the first shot we don't see the door touching the strike plate or frame, and in the second shot you don't actually see anyone locking it, but I am pretty sure that there are many doors that behave in a similar way in real life.