you go to the eliet 4 and finesh the game.
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The badges are just,there for you to be able to use HMS outside of battle and for high levels of Pokemon to listen to you depending on the badge...
its actually the 8th gym so you need 7 badges and then it will unlock
The gym in Viridian is actually the last gym. You'll unlock it after getting all the other badges. You should continue to Pewter through Viridian Forest.
you get all 8 gym badges
No, you can trade with no badges at all.