You get it by beating Giovanni at the Viridian City gym, which you can't do until you have the other 7 badges.
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After defeating Blaine at Cinnibar Gym, you must return to Viridian City. The gym will now be open, and after defeating the leader, you'll get the Earthbadge.
ummm well you go to viridian city and fight Giovanni.....he gives you the earthbadge and the TM earthquake you can then challenge your rival on the way to the Pokemon league
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.