No, eggs are only available in the generation one games, there are by breeding Pokemon in two and up though.
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yes you can transfer a egg by link cable in the versions gold/silver/crystal. but u cant trade eggs to red/blue/yellow/green.
No. They were made after Pokemon Yellow.
No, you cannot get Lugia on Pokemon Yellow.
Yes you can trade Pokémon between two Pokémon Yellow games.
You can get rare candies in pokemon yellow