During my turn, I place a building which produces resources or provides coins. Can I then use those resources / coins to pay the participation cost of a Great Project? Could I use those coins to trade with another player for a resource needed as part of the participation cost?
The language used to describe how to participate in Great Projects indicates when you need to pay the cost. Unlike monetary losses, which are "always resolved at the end of a turn, after players have played their cards, possibly having paid for trade and/or obtained money," participation is an additional cost at the time of playing your card. From the rule book on page 6 (emphasis mine):
From now on, when a player constructs a building of the same color as the Great Project card in play (Brown, Blue, Red, Yellow, Green, or Purple), that player has the option to participate in its construction. To do that, the player must pay the participation cost in addition to the cost of their building. In return, the player takes a Participation token which they place on their Wonder board.
The decision to participate happens as you are paying costs, which must occur before gaining resources from your played card. If you don't have the resources at the time you are revealing your card, you can't participate in the Great Project.