According to 2K staff on their forums, it is not possible to gain trophies/achievements in the New Game+ mode of BioShock.
As for what carries over:
New Game Plus allows players the ability to start a new game with the character data (weapons, inventory, research, etc.) carried over from the end of the game (if they beat the game after the patch was installed) or from a saved game of their choosing (if they beat the game before the patch was installed).
When using New Game Plus, you will gain access to all of the tonics, plasmids, and weapons you have acquired previously in the game. However, you will not be able to access these goodies until you reach Medical, when you will receive all the weapons you had previously uncovered, and three plasmids. The remainder of your plasmids and tonics will be waiting for you at the first Gene Bank you access.