You have to have and use the HM 08 Rock Climb, which you obtain from Professor Oak after obtaining all 16 Kanto and Johto gym badges.
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The magmorizer is on Cinnabar Island. You must defeat Blue first before being able to get to the top. Professor Oak will give you HM Rock Climb to get up to the top of Cinnabar, where somewhere on that island is the magmorizer.
To find Lugia enter the Whirl Island Caves at the top-right island, then it should be pretty easy from there (if you're stuck use youtube or similar.)
Nope. But you can in soul silver or heart gold. (At the top of mount silver)
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you battle and encounter your rival at the top of mt.silver.