Gate key:
You need the Fire Sword. Near the beginning is a room with a red "pig". In that room is a door with a ice block. Melt it to get the flood gate key. Then, go to the room with big skeletons and unlock that door. Though there is the key that you are looking for.

Unlimited fairy experience:
Use the Light Fairy to see, when in the ruins under the city. After getting the fairy, return to the first save location, then go right. Enter the first door from the save location. Go through the little room and go to the next door. Look out for the robots in this room, as the shield will not block their attacks, which do a lot of damage. Go after them at an angle. Use the fairy in the room after the robots. You should now be able to see a small blue square on the floor. Use the square to turn on the lights, then get on the small flying square. It allows you to travel over the electric floor without harm. There will be a few enemies in the room that will always have a ball that increases your fairy's experience by 50. This may be repeated as many times as needed.