You can find Shiny Pokémon in all games except in Red, Blue and Yellow.
The encountering rate depends on each game but they are pretty rare.
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Yes the same way you get shiny Pokemon in the other Pokemon games is how you get shiny Pokemon in heartgold. There is a free shiny Pokemon in the lake of rage.
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.
They all have shiny Pokemon; just the chance of you encountering one is very low.
In the 2nd generation games (Gold/Silver/Crystal), the offspring will have a 1/64 chance of being shiny. In later generation games, the chance of getting a shiny offspring is the same as encountering a wild shiny Pokemon.
Yes it can.