If a player builds a new SD in a system, thus, activating it, may that player build in that activated system, in that SD later on through the secondary action of the Production SC?

That's in the same game round, lets say, player A takes action, activates system and builds SD, then player B (next in order) plays Production SC.

No, he may not produce from a SD built in the same round.

Yeah, this is one of the few times where production doesn't offer the workaround of building in an activated system. One of the solid rules of the game that is only broken if flat out stated in a card or ability is that a space dock cannot build the same ROUND that it is built. ROUND, not TURN.