From bulbapedia.bulbagarden.net:
Gyarados's name may be derived from a combination of 虐殺 gyakusatsu massacre/slaughter, and 逆境 gyakkyō, hardship/adversity. Both words relate to the creature's violent nature and the hardship it experienced before evolving. The end of Gyarados' name, ドス dosu, is a Japanese onomatopoeic word representing the piercing of flesh, once again owing to Gyarados' violent nature. Alternatively, the dos may just indicate that it's the second in its evolutionary line.
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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.
Yes, the "Red Gyarados" is actually just a shiny Gyarados.
Gyarados does not evolve into Lapras.
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.
another gyarados, or a magikarp of the opposite gender. (it should work, I've never tried breeding Gyarados)