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Trap tokens let you place 1 or 2 cards face-down into an Offering instead of face-up, and neither player may look at face-down cards in the Offering; if you take an Offering with face-down cards you put them into your Selection area still face-down and only reveal them in the Planning phase (Rulebook, p.3).

Treasure cards must be recycled during the Planning phase and give 2 resources when recycled, while Calamity cards cannot be recycled — you must add them immediately to your Duchy and they cost −4 VP at game end (Rulebook, p.4).

Yes — when a card's construction completes you add it to your Duchy and it will be able to produce later in that same round (for subsequent Production steps); likewise, recycling during Planning that immediately completes another card also lets that card produce that round (Rulebook, pp.4–5).

Whenever you have 5 resource cubes in your Duchy's Alchemy area you immediately discard them to gain 1 Krystallium cube; Krystallium can be kept indefinitely and used as any Resource cube or to fill spaces that specifically require Krystallium, but it cannot be used as a Soldier token (Rulebook, p.6).

No — except for Krystallium, you cannot save Resource cubes from round to round, and you cannot move Resource cubes once placed on a card under construction or in the Alchemy area (Rulebook, p.5).

After each Production step you announce your production total; if you produced more than your opponent you gain the Supremacy bonus, which lets you take 1 Soldier from your Training area or place 1 Soldier there if none are present, and Soldier tokens are kept until used by module effects (Rulebook, p.5).