Why didn’t the trap work

There are two types are armor, Gunner and Blademaster. Gunner can only be worn by hunters using a Bow or Bowgun. Blademaster is for hunters using any other weapon. The hunter type of every piece of equipment is displayed in the items details, on the second screen (accessible with the "R" button). When you're in the blacksmith you can change the armor type displayed by pressing the "-" button. The items currently displayed are shown at the bottom of the list, next to "Current list".

Helmets are the only piece of armour lacking a type, and therefore can be worn by any hunter, regardless of which weapon they're using. However, every Helmet is intended to go along with a specific Gunner or Blademaster armour set. The Gunner version of a Helmet will normally have different skills than the Blademaster armour, and will always have less armour. Looking at the defense is an easy way to tell which set the Helmet is intended for, since the helmet with lower armour goes with the Gunner set, and the higher armour helmet goes with the Blademaster set.

You can only deliver eggs to the red box when it's a quest objective.

If you get an egg at some other time, you can bring it back to the Veggie Elder at the camp. He'll give you a Jumbo Pearl for a regular egg, and a Wonderful Beak for a golden one.