A few things. First, the film is barely 2 hours long with credits. You could do a lot with 45 more minutes.
Hmm. It really depends where the story is going, or where you want it to. You have complained endlessly that the film just sets up questions without answering them, which is a fair point. But can you envision a successful answer to all of the questions over the origin of our species, the history of the Engineers, our interrelation with them, how the Xenomorphs fit in etc, in 45 minutes? You might be able to compress something like a conclusion in there, but I would hardly be structurally pleasing, and the extra time would have to largely be exposition.
But even then, if you had to drop stuff? You can drop the cave discovery.
One of the nicer sequences in the film, which broadened the scope somewhat.
We get the exact same information less then ten minutes later.
In a bit of clumsy exposition. That was one of the weaker scenes. I would rather keep the cave exploration and just have the team know what they are doing already.
Get rid of the two idiots
Fifield was a crappy stock character, so I am inclined to agree, but more was needed than Shaw's monstrous pregnancy to show the importance of the ooze and its uses. The loss of that whole section would remove much of the horror/suspense from the film. I would agree that it could have been better, but I don't think you could just pull it out without an even less satisfactory film. After all, we don't like are questions going unanswered, huh?
or simply have them disappear and have them found messed up.
I have the inkling of a suspicion that, had they done that, you would have called it "lazy writing".
For me, the storm was the most visually pleasing scene in the whole film, which gave a lot of physicality and believability to this distant planet. You could cut it, but then there would be no reason for anyone to get stuck in the temple with the ooze, and a lot of visual flair would be sacrificed for little gain.
and the attack by corpse.
Not a good scene but, again, the ooze is evidently key to the whole mythos, and you would just be leaving yet more of the dreaded "unanswered questions" if you didn't trouble to explore what it is or what it does. It's another scene that I wish was better, but not necessarily absent.
What you suggest certainly makes for a lean film, but it is one shorn of much of its imagery or its already limited action. Basically, the team wake up and a presentation tells them what's going on. They land the ship and go to the temple. Nothing happens and they come back. David poisons Holloway, Shaw and Holloway have sex. Next day, they go back to the temple again. Holloway gets ill, and they go back. Flamethrower scene. Nasty pregnancy stuff. Wayland is woken up, and they all go to the temple again. The Kurgan kills all but Shaw, followed by the same denouement we all saw.
That structure as it stands doesn't really work, because there is far too much to-and-fro from ship to temple. You could argue that it could be padded with whatever the next film intends to tell us, but I think it's safe to assume that the context needs to change to the Engineer's planet for that stuff to happen. I certainly doubt it could happen before the Kurgan awakes. And then you would have a lot of leaden exposition for the audience to contend with.
I know you really don't like questions without answers, but you wouldn't be very interested in a three hour question and answer session, would you?