How David knows how to open the 1st door?

He opened the door the old fashion way...he hacked the system.

He spent 2 years studying common roots of ancient human languages. He says he is confident in his ability to converse with the Engineers. Extrapolating that, he could read their writing. Hence opening the doors. Hope that helps :)

I think that after he touched that slimy substance on the wall before opening the door and setting off the hologram. ( the stuff that when he looks at it he see's green stuff like a dna code) he somehow understands what the engineers were after and facilitates it. Now i know this is a stretch and the reason i believe this is in the way david is treated throughout the entire movie. He is just not considered to be even close to a human level and despises the humans for this, comparing himself closer to an engineer. This how he is able to understand the the symbols and eventually speak it.

Communication David 8 is fluent in all known languages through a dialectic implant and can infer the linguistic components of entirely new languages if encountered. Communication between humans and David 8 should feel entirely fluid and natural. (David 8 specs)