Long time RPGer, first time with Pathfinder.
Welcome!
1. Dragon Stance is "Open." The Open Trait says: "You can use an action with the open trait only if you haven’t used an action with the attack or open trait yet this turn."
Does this mean it basically has to be the first thing you do if you want to attack with it this turn? (I.e. no moving from stance to stance or attacking first then moving to a stance).
Not quite; the examples you gave are forbidden, but you could still draw a weapon and/or move into position, or any other non-attack non-open actions, before taking a stance.
2. Once in Dragon Stance, can you do Flurry of Blows? (It's Unarmed, but the Monk Weapons implies that stances--like Crane--require you to make a "single, specific attack.")
Yes, you can, and in particular you can flurry the "dragon tail" leg attacks.
What Monastic Weaponry is saying is that since Crane Stance gives you a specific unarmed attack (the crane wing attack), you can only do it that way (unarmed), not with a weapon. The same would hold for Dragon Stance and the dragon tail attack; you can flurry it, but you can't do it with a staff, even if you have Monastic Weaponry, only with your leg.
However, you could go into Dragon Stance and make attacks other than dragon tail attacks, if you wanted the other benefits of the stance (like ignoring difficult terrain). In that case those attacks could be with weapons, and you could even flurry with a monk weapon if you have Monastic Weaponry. This is not the case for Crane Stance because that contains a "the only Strikes you can make are crane wing attacks" clause.
Hope that helped.