HeroQuest

No — you cannot pass over monsters or share a square with a monster (except on the stairs or if you are in a pit trap); monsters block movement. (Rulebook, p.7)

Yes — looking and opening doors are not one of the six actions and may be done while you are moving, so you may move adjacent, ask Zargon to open the door, and continue your movement. However, you may not move part way, perform a (six) action, then finish moving. (Rulebook, pp.6–7)

Yes — searching for treasure without first searching for traps can spring chest/furniture traps (and moving onto hidden pit, spear or falling-block squares also automatically springs those traps). Be warned that many treasure cards also contain wandering monsters or hazards. (Rulebook, pp.8–10)

No — monsters do not spring hidden traps, do not search for them, and cannot disarm them. (Rulebook, p.11)

Most heroes need a tool kit and must move onto the trap square and roll 1 combat die (shield = disarmed, skull = sprung), but the dwarf never needs a tool kit and disarms on a 1‑die roll (anything except a black shield disarms; a black shield means the dwarf sprung the trap). (Rulebook, pp.10–11)

You can avert death only by immediately drinking a healing potion yourself or, if you are a spellcaster who has not yet acted, casting a healing spell on yourself; once your Body Points have reached 0 you cannot be saved by a fellow hero’s spell or potion. (Rulebook, p.11; see also p.12)