If I use Bran to reveal Search and Detain Round 1 in the action window before the initial plots are revealed, is anyone even first player yet? Similarly, when does first player reset? If my opponent (joust) is First Player Round1, does that status get reset at the end of the round? Or does it last until the next plot is revealed?

You actually should determine first player randomly before each game. You don't because it usually isn't important. Good catch there though. A pretty strong first turn preplot action. Also you stay the first player until a new plot is revealed and a new first player is determined.

The rules do tell you to randomly choose a first player for setup. Technically, the first player in setup is supposed to announce House/agenda first (before the next player does). The first player is also supposed to place setup cards face-down first (essentially letting the other players see how many cards they drop), even though everyone reveals at the same time. These details are close to meaningless for most players, so almost no one ever does this, but it is in the rules.

As Staton says, when someone is named first player, they remain first player until a new one is chosen. There is always a first player.

By the way, if you do use this "setup-Bran-into-preplot-Search and Detain" trick, I'd suggest following up with Rule By Decree as your standard plot in Round 1.

I remember a few years ago there was some discussion about this because there were a few people playing a decktype that ran the Alliance agenda and a whole bunch of traitors. In order to work optimally, they'd need to go second, see what house their opponent was playing, then name that house as their Alliance house. Thus, they'd avoid the gold penalty for the traitors that wrecked that opponent specifically.

I don't think the deck was super competitive, but it was a fun idea, IMO.

(of course, most Traitors in the LCG environment are House X only, so I don't think that's really feasible anymore. This was in the CCG days.)