Yes you can. Though, it is very rare.
One way to improve your chances of getting a shiny Pokemon by hatching an egg is by breeding a Pokemon that is of a different nationality. This meaning you must be traded a Pokemon from a different game. (i.e. a Japanese Pokemon onto an English game)
This will increase your chances by 75%.
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With breeding 2 normal Pokemon, the chances of getting a shiny Pokemon are 1 in 8,192. By breeding 2 Pokemon from different countries, the chances of the offspring being shiny is 1 in 2028. With a shiny female Pokemon and breeding, the chances of that Pokemon being shiny is 1 in 64. so, yes, it is possible, but it takes forever.
Yes, shiny pokemon can be breed. However, breeding shiny pokemon does not increase the chances of having shiny offspring, except for in Generation II games.
yes
A golden Margikarp is a shiny margikarp, and as with other shiny Pokemon, there is only a 1/1892 chance of getting a Shiny(golden) magikarp. Breeding a Shiny Gyarados or a Shiny margikarp with another shiny Pokemon THAT CAN BREED WITH GYARADOS/MARGIKARP may have a higher chance of breeding a shiny magikarp. You can also trade the shiny gyarados from Pokemon Gold, Silver, and Crystal version.
i think there is a way , but you have to have a shiny Pokemon to get it in shiny, or i don't know if you can put a shiny ditto so the Pokemon can come out shiny.Maybe you can put two shinys in the daycare and it will be a shiny Pokemon in the egg.I hope these ideas work:) I've tried breeding Shiny with Shiny, but it didn't work. It's just luck; you'll never know.