With The Advanced Gunnery the 2nd player has an advantage as the can target the same hull section twice.

Is this determined at the start of the game and applies throughout or does it depend on who is the 2nd player each round?

Second player stays as second player the whole game.

Like each round, first player is the first player on the first round.

Alternating initiative is only used when playing without objectives or a round limit. In a standard game, one player has initiative (i.e. is the first player) the entire game. This is why the objectives all favor the second player.