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I think that cave is also in the form of the leaked photos we saw a few months ago that had the urns and tablets as well as some creatures. So, this could David's lab so to speak and his home. I'm under the impression this is where he brings the surviving members of the crew that didn't die when the Dropship blew up. David comes to their rescues, possibly killing the two Neomorphs that had gotten loose. David then brings them here where he lures Crudup to the egg room. Crudup then becomes a host for David and possibly the first for the facehugger experiment and as we see in the photo above, that lump is Crudup who somehow awakes after and makes it back to the ship where he gives birth to the "protomorph" and we see it briefly in the shower scene at the end of the trailer. 

@David007

Interesting prediction! The film may very well go that route, quite possible! Perhaps Shaw was the first 'failed' host?

I'm gonna go out on a limb and say that David 8 is not going to be the heavy.  It looks like Walter is the one talking to "Weyland" in the white room.

Weyland was the true villain of Prometheus. He was pulling David 8's strings and David implied that he hated weyland for what he was being told to do.  

Until Weyland dies, David 8 had no free will. Once he did, he helped Shaw. He even tried to protect her from a dangerous choice to go to the engineer home world.  

Anyone interacting with Weyland in any form is clearly not on the right side of crew survival. Whichever Android talks to Weyland is the clear baddie. David unleashed the black goo for reasons unknown on the engineers there. That created the neomorphs.  The "protoXenos" come after the black goo outbreak. (My theory only)

@Myrddin365

I assumed the Engineers were long gone by the time Shaw and David arrived on Paradise.  But if the Engineers were still there when they arrived, I doubt they would let David get close enough to release the black goo on them.  I think the Engineers just lost control of the black goo.  

And I'm also not sure where you got that David hated Weyland for what he was making him do.  I think David's actions were out of sheer curiosity, in Prometheus and Covenant.  

The way I see the white room scene is...  It took place before the Prometheus mission.  It is a conversation between Weyland and David about gods and mankind, etc. This is basically laying the groundwork for the whole reasoning behind the Prometheus mission.  It is also the point where David's curiosity started, hence his actions afterward.  

Seeing as how Weyland is dead by the time Walter comes into the picture, I think the white room scene has to be David.  

But obviously we can't say for sure at the moment.  

Walter is the brunette. There is an unambiguous picture of him sitting at the piano. David explicitly stated "doesn't everyone want their parents dead." The simplest explanation is because he wanted his parent dead. Which implies hatred of that parent.

Weyland's holograms knock on the door of synchronicity.  It follows that he would pursue that avenue as well.  So he has holograms of himself stored to communicate EXACTLY what he wants to say should certain conditions arise. They have.

I think that David tricks the crew of the covenant into thinking that he is in fact Walter. In Prometheus we saw David putting peroxide in his hair to become blond. It is unlikely that he would have continued this since the events of Prometheus and will thus have darker hair like Walter. All he would have to do then is get Walter out of the way and take his place, convincing the trusting crew to follow him to the cave.

@Myrddin365

I think David was referring to Vickers when talking about the dead parents.  But you could be right.  He might very well have wanted Weyland dead also.  But I don't think he ever lost his appetite for curiosity through all of this.  

You're also right in that the white room scene could be a hologram.  I still think he was alive during it, and that it happened before the Prometheus mission.  And hair color aside, I still think it is David.  

But hey, that's the fun in this, having different theories.  We will find out soon enough!

Also, I don't know if this qualifies as confirmation, but AVP Galaxy talked about this scene way back in October and said it is a young Weyland and David and that it did take place before the Prometheus mission. 

Heres a link to a website documenting it: 

http://www.slashfilm.com/alien-covenant-details-revealed/

Remember,david colored her hair in prometheus.