1. Only if you bring it to a new game, for example, using the Sailboat to start a new game with your items.
2. The milestone rewards are one time, but you will get better (and more) starting items the more milestones you have unlocked when starting a new game.
3. Traps start out pretty weak. Their damage and chance at triggering are based on your Trapping skill. If you get this near 50%, your success rate will be 100%, with extra damage occurring each 10%. Full range is 3-16 damage currently and counts as pure damage, meaning it will go through the monster’s defense and resistances.
