The only known way of raising your chances of getting one are encountering more Pokemon in the wild or using a GameShark/Codebreaker code.
They can produce a shiny if bred, but they wont always. The odds are much better if you catch and breed. It goes from 1 in 8192 to something like 1 in 64. Also, If an egg is bred from two Pokémon which were created in games of a different language from one another (either by capture or by breeding), the resulting egg is four times as likely to be shiny when it is generated. (bulbepedia)
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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.
Very rare
1/8192 is the appearance rate for any shiny pokemon. The odds are no different with legendary pokemon.
No, the egg will not be shiny. However, it is possible that the egg will hatch into a shiny Pokemon, but the chances of getting a shiny Pokemon from the egg are not any higher than they would be if the ditto were not shiny.
Getting a shiny Pokemon is dependent on luck. The chances of encountering a shiny Pokemon is 1/8192, so it may take a long time to get a shiny Abra.