Spoilers:
I think that whether the opening scene indicates that Conner is dead by the time our heroes arrive depends on its relation chronologically to the other scenes. Clearly, there was an elapse of time between the opening scene and the rest of the scenes. The question is, has time passed, or is time passing? If the opening scene is the past, then yes, Connor is definitely dead, and Garret met up twice with a fake Connor on the ship. On the other hand, if the opening scene shows a fragment of the future, then Connor may still be alive, and the Connor Garrett met is either the original Connor, the fake Connor, or both.
Speaking of other horror media, the carved stones in the lab and the vault remind me of the Red Marker from the Dead Space video games.