Dungeon Lords

Traps are revealed and resolve in the Trap Step at the start of the battle (before spells and monsters); if a Trap Card has a cost you must pay it (in rooms you must pay 1 Gold) and you may not choose a trap you cannot pay for (Rulebook p.4, p.19).

Adventurer Tiles are assigned in the Adventurers Phase and placed from the entrance outward (new warriors push to the front); the paladin appears immediately when your Evil Counter reaches the paladin space and is placed at your entrance as a special front-line adventurer (Rulebook p.17, p.8).

You cannot use an action unless you pay its listed cost; if you don't pay (or can't) you simply don't perform the action and move your minion to the Orders Card (Rulebook p.13). Only when an action explicitly lists an Evil/Reputation cost do you pay by moving up the Evilometer; if you cannot move up you cannot pay that cost (Rulebook p.13, p.22).

You take the newly recruited Imp Figures by paying the listed Food/Gold, but they cannot be used for Dig Tunnels or Mine Gold that same round (they become available for Production and later actions) (Rulebook p.14, p.11).

Yes — conquered tunnels and rooms remain conquered and cannot be repaired for the Second Year (Rulebook p.20). Scoring is done with the Overview Card: points for unconquered rooms, monsters, prisoners (paladins worth more), penalties for conquered tiles and Dead Letter Office, plus bonus room scoring and titles (Rulebook p.21, p.24).