I can't seem to find a rule about the terrain and Objective decks containing only 1 copy of each card. I have 2 core sets, so can I put multiple copies of the same card into either terrain or objective decks?

Terrain yes, that is legal for OP even, capacity limit permitting. Objectives I'd rule no, as they are being selected round by round across the board, so we don't have a player controlled way to affect objective/deployment.