The cards are taken secretly from the discard, and then put into play simultaneously
The quote from the Wonder Pack rules should be taken literally. When they use the term, "resolution order" they are specifically speaking of what order to resolve the actions in the previous conditional: (I.e. multiple players taking cards from the discard pile.)
If multiple players take cards from the discard pile during a given turn, the resolution order is the following: Halikarnassos, The Great Wall, Manneken Pis, Salomon and finally, Courtesan’s Guild
Cards are always played simultaneously, so some way to resolve conflicts between multiple players taking conflicting actions is necessary. Take for example the original rules (page 5, 8):
Overview of an Age - [...] Each Age is made up of 6 game turns. During each turn the players put into play a single card, simultaneously.
The Mausoleum of Halicarnassus - the player can look through all of the cards discarded since the beginning of the game (those discarded for coins as well as those discarded at the end of an age), pick one, and build it for free.
Clarification : this special ability takes place at the end of the turn in which this stage is built. If players discard cards this turn (as in the case of the 6th turn of an age), the player can also choose from among them.
The same rules text is in the Leaders Expansion (page 4, 7):
Recruitment Phase - During this phase, each player will play one of their Leader cards. The cards are chosen secretly by each player, then revealed simultaneously to be played.
Solomon - Once Solomon enters play, the player can choose an Age card from the discard pile and put it in play for free.
The Leaders rules also only notes that Halicarnassus chooses its card before Solomon, not that it plays the card before Solomon.
Q: What happens if Solomon enters play in the same turn in which Halicarnassus builds a Wonder stage?
A: Halicarnassus chooses its card from the discard pile before Solomon.