Yes, but the chances are exactly the same as usual. The only way to increase your chance of hatching a shiny is to have a foreign parent, which halves the shiny chance I think. Hatching a shiny has the same odds of finding a wild shiny.
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Get a shiny Magikarp. A Red Gyarados is a shiny Gyarados.
Yes, the "Red Gyarados" is actually just a shiny Gyarados.
The red Gyarados is simply a shiny Gyarados. This means the red Gyarados is not any stronger or weaker than other Gyarados. Gyarados in general is considered to be a very powerful Pokemon for how easy it is to get.
technically, yes. There is an 8462 to one chance of finding a shiny pokemon. If you found a shiny gyarados, it WOULD be red. You could also evolve a shiny magicarp(which, by the way, is gold). No, there is no guarantee on this, and there is no guaranteed chance of finding a red gyarados.
No, Gyarados and red Gyarados are basically the same having only one key difference; red Gyarados is a "shiny" Pokemon which changes its color and makes it rare to find in normal gameplay.