Do the higher diffculty quests in the summer event drop more materials than the lower difficulty ones?

Pretty much. The higher the difficulty the more mats dropped per mob. If you're looking to clear the shop the most AP efficient way to do it is use groups with proper passives and event CEs and just keep on doing it on annihilation difficulty. In fact it's also more ap efficient to skip the 20 AP ones. All this is assuming you can field a team or teams capable of clearing it on Annihilation difficulty