So this happened last night and it caused some real issues for us. One person became pretty upset and I'm hoping someone can clarify.
My friend played the market. Drew a card and it was the thrown room. Can they then apply the thrown room to the market card which they played that led to them getting the thrown room into their hand? Seems like they can't but I didn't see anything that says that the thrown room has to be declared before you start playing a card. The Market allows the user to draw a card and then do an action so the card hasn't resolved at the time that the thrown room was drawn.
The bolded part is probably contributing to your confusion. It's quite a common misconception.
"+x actions" just adds x to your "action counter", which is just an imaginary number floating around in space somewhere. It
doesn't mean "play x actions now". You resolve the whole Market, all the way up to the end, before thinking about playing any more actions. Then you check your action counter: "Oh look, I still have 3 actions left on my counter, I can play up to 3 more actions this turn." And so forth.
The only time when you play other action cards during the resolution of an action card is in things like Throne Room that say "play an action from your hand", or things like that. In that situation, the Throne Room is not fully resolved until the action you play with it is done.