The first player chooses the order.

The active player marshals his cards face up for everyone to see?

When you marshall a card, put a card into play or play event, you do so face up (except if you're marshalling a card into shadows).

And the marshaling phase order is indeed determined by the player who won initiative?

mbm said:

When a player chooses the First Player during the plot phase, that player acts first in both the Marshalling and Challenges phases. Turn order then proceeds to the left. So the winner of initiative only really chooses turn order by choosing the First Player - with everything else determined by table order. So the First Player Marshals first, then the player to his/her left. and so on. Same in Challenges - the First Player does all of their attacking first, then the player to his/her left, and so on.

It can be a little confusing because of another game mechanic. The First Player has the "power" to decide the order in which conflicting passive effects - ie, passive effects that would initiate at the same time, but can't; like when 2 or more players use "when revealed" plot cards at the same time. When the First Player decides the order of such effects, s/he does not need to follow turn order and can say the passive effects happen in whatever order they want.

It's easy to over-generalize this power. A lot of people do, thinking the First Player can decide the order of pretty much everything. As an extreme example, people have asked in the past if the First Player gets to choose the order in which players use Responses to the same thing (like winning/losing a challenge). So it's easy to see where turn-order vs. First Player decision has been a source of confusion when learning the game. But the "conflicting passives" power is a lot more limited in practice than it looks on paper. There are a lot fewer timing conflicts in the game than first appears.

(Heck of a lot more information than you probably wanted, but hopefully useful in the future.)