Yes, blue-eyes shining dragon can destroy a rainbow dragon.
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Yes it does. Lightning Vortex does not target. Blue-Eyes Shining Dragon's protective effect works only against cards that target, like Exiled Force.
No, if you look at Blue-Eyes Shining Dragon's text, it says it cannot be special summoned, except by tributing a Blue-Eyes Ultimate Dragon from your side of the field.There is simply no way around this. It is the only way it can be summoned.
Blue-Eyes Shining Dragon is only protected from things that target him. So you can use the many non-targetting cards against it. You can negate the summon with Royal Oppression or Solemn Judgment, you can destroy it with Bottomless Trap Hole, Smashing Ground, remove it with DD Warrior Lady, etc.Blue-Eyes Shining Dragon is only able to negate things things that target it.It is therefore vulnerable to the following non-targetting effects: Mirror Force, Lightning Vortex, Smashing Ground, the DD monsters, Neospacian Grand Mole, Judgment Dragon, something powered up by Honest, to name just a few.When it uses its Quick negation effect, Divine Wrath or GB War Chariot could be chained. Blue Eyes Shining Dragon could not then activate again to stop those Counter Traps because they are Spell Speed 3, and BESD's negation effect is Spell Speed 2.
To easily defeat Blue-Eyes Shining Dragon, use cards that don't target a specific monster on the field (Dark Hole, Mirror Force, Raigeki, etc.), since that monster's effect states that it can't be defeated by card effects that target this specific card.