The funny thing is for me I loved the original opening, would have moved the Isle Of Skye into one of the dreams David is watching but the real crap starts with the briefing scene.
If you cut anything that scene has to go.
Make it so the crew know where they are going and that Weyland is onboard already.
The less of Guy Guy Pearce's awful Halloween cake makeup the better.
I'd drop the angle about the alien buildings being a weapons lab too.
The opening establishes that this liquid does a job and they have to make the stuff somehow so why not have the alien in the 1979 film the main ingredient in their ritual DNA solvent/ re-compiler?
That explains why they venerate the alien.
Without it they wouldn't be able to spread life.
That way the ship in 1979 is still old it's shipping the cargo presumably to the place in this film and something went wrong and killed the goo makers.
Two other options spring to mind.
We see lots of buildings so it would make sense if Milburn and Fifield the geologist/cartographer go to the next building (not freak out and run away) to prepare the ground for further investigation.
They get lost in a place that isn't mapped yet during the storm and then return to the other building because the corridors are moving around bringing them back to the ampule room.
I'd try and cut out the reanimation of the head, it's really silly.
I'd trim as much stuff showing them to be idiots as possible (it's scarier when plausibly intelligent people get killed).
The other bit I would be tempted to lose even though it's creepy is the fish in the eye bit.
It makes more sense that Holloway begins to visibly sicken away from the ship.
If he suspected something was up he would at least have told the medical officer and one hopes Shaw too.
I'd try and edit the Fifield mutant attack (the CGI one if possible) into the sequence where Holloway is killed.
They return and it gets on the roof of the bus. It creates merry hell and kills people Shaw crushes it under the wheels of the bus and Vickers torches Holloway.
I'd have the squid baby die in the medpod.
The engineer is already infected with an alien (that is why is in hypersleep) and he is as much in a rage because David has killed him by thawing him out as in his question.
If possible I would reverse the impact of the ships.
So the Prometheus is trying to head home and the engineer rams the Prometheus to maintain quarantine.
I'd remove Shaw's assumption that the Engineers lost faith in us.
There is no evidence for it.
It's pure assumption as is the assumption that they made us.
I'd leave it as an open possibility but remove it as a main motivation.
Weyland may believe it or at least believe they are advanced enough to extend his life but having it just taken for granted weakens the film as does the 100% DNA match (I'd lose the results entirely as they are scientifically meaningless gibberish).
In the challis scene we are never sure which world it is.
The implication that it's pre-historic Earth should remain that, an implication.
I'd try to remove that annoying Trekish score motiff too.
Nice to hear the Alien 3 Fox Fanfare again.
While the TED talk does lead well into the Prometheus in space I don't get your antipathy towards the original opening. It's probably the only thing in the film that really deserves preservation.
I can see Engineer scene, TED talk, Prometheus, David doing his thing (no need for the 'Happy Birthday David' material unless it was shown on the same screen as Lawrence Of Arabia). Have the Skye cave as Shaw's or Holloway's dream and the Quiet Eye as Weyland's or Vickers' dream.
I don't know if it's possible but maybe David can access the Engineer's memories directly when he is in stasis like he does the crew, that way the challis scene can be kept but your intension of not revealing the Engineers until later can be maintained.
That's just my perspective though.
I'm looking forward to seeing an alternative take on this film which really disappointed me.