If both players have either Ig or Luke with their respective CC in hand, is there a time window after a figure resolves his activation to play the card? I.e. the player with initiative has to play it first?

Both are after a (figure) activation, so the player with initiative has to make the decision first. If the player with initiative decides not to play, the player cannot change his mind later. However, the other player(s) cannot rush the player to miss the opportunity.

(I assume after an activation is strictly after at the end of an activation if there are such abilities in play, even if they happen at the same time in real-time.)