You go to Victory Road
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you get all 8 gym badges
First you need to get all 8 badges and past through to the cave that close to it.
No, you can trade with no badges at all.
You are in jhoto first then when you have all the 8 gym badges you can go to kanto and get 8 more badges which will be all 16 badges then you can challenge the pokemon league. Hope that helped.
You can get Rock Climb by first getting all 16 badges. Rock climb cannot be found as an HM in the ground somewhere, but from a person. Professor Oak will give you it as a prize for getting all 16 badges.