1) Your opponent summons Jinzo, can you chain to it with Bottomless Trap Hole?
No, Jinzo's effect applies immediately after successfully being summoned.
2) Your opponent has an Exiled Force on his side of the field, and you have a Junk Synchron and 2 Doom Caliber Knights. He uses Exiled Force's effect to destroy your Junk Synchron, what happens?
Doom Attempts to negate Exiled, but the other Doom negates Doom, therefore both DCK's die, Exiled's effect resolves.
3) Can you summon Blue Eyes Shining Dragon even if you don't have three Blue-Eyes in your graveyard?
Sure, why not? xD
4) Can Caius the Shadow Monarch target himself?
Yes, that's called a Caius nuke, when you need to deal 1000 damage to win, but don't want to risk attacking.
5) When your opponents summons a Breaker the Magical Warrior, and you chain to it with Bottomless Trap Hole, can he us priority to destroy an S/T?
No, Breaker's Priority is to Gain a Spell Counter, not to destroy S/Ts