Moves can only get passed down by the parents if they are Egg moves, and Magikarp doesn't have any Egg moves so no, it will not have the same moves as your Gyarados.
The nature of a Pokémon is completely random unless you go to catch a Ditto or Magikarp with the right nature and give the female Magikarp/Gyarados or a Ditto an everstone to hold. The everstone makes it so there's a 50% chance of the female/ditto passing down its nature to the hatchlings. (This was changed in BW2 to make it so that the male Pokémon can also pass down natures, and its a 100% chance of it getting passed down)
There are two things you can do:
-Keep hatching Magikarps and hope you hatch one with the right nature, it shouldn't take too long.
-Catch a female Magikarp/Gyarados or a Ditto with the right nature and give it an everstone to hold, so then you can have as many good-natured Karps as you want!
Sources: Magikarp and Everstone.