When you encounter the elf on the exploration trail, and it says that it requires a "hat" to bargain and I have one, how do I give it to him? And the red gift boxes - what do I do with those? It says give to elf, but how?

You need candy, gift, and a hat to give to the elf. You can't just have one you need all three. And it does it automatically when you have all three.