1. Yes. "The Detective also has the option of using the "gun's blazing" mode once per game, which allows them to kill any one player regardless of whether or not that player has been investigated".
2. After the voting. It can be bad enough that the narrator isn't around for when someone is supposed to get jailed, and having a second role that is so specifically time dependent can mess up the game. But that's just my opinion.
3. Both killings occur.
4. I usually play this one that if the D sends his request before he's jailed, it stands. While the role doesn't get their info until after the voting, his duty is one that is performed during the day time and if he does it while he's still within fair rules, it's fine. He phones in his investigation in the morning, gets jailed, and upon release there's someone with an envelope waiting for him that has his answer.