As a Trigger, Tiger Dragon's effect will go on the chain first before anything else can be used, assuming the summon was not negated. Cards like Bottomless Trap Hole will be chained to it.
Even though BTH resolves first and destroys the Tiger Dragon, this does not stop his effect from resolving and destroying the two targets.
No. No part of Bottomless Trap Hole negates effects. However a monster's continuous effect requires it to remain face-up on the field. In those cases, the monster effect will cease to apply as the monster is gone.
Torrential Tribute and Bottomless Trap Hole can only be used in response to a monster's summon. Attacking a monster does not summon it (note that if the monster is face down, this is a Flip and not a Flip Summon).
Yes, you can, it is a special summon and can be responded to as normal. However it is more than likely that the monster's controller will simply tribute it to negate and destroy Bottomless Trap Hole, then resummon the dragon in the same end phase.
The only times a monster is considered 'destroyed' are if the card that removed it from the field says it 'destroyed' it, or if it was destroyed during Damage Calculation with a stronger monster. So Bottomless Trap Hole removes from game, but also destroys. However Dimensional Prison only removes from game, this is not a 'destroy' effect.
you sacrifice a spellcaster on your side of the field and special summon a spellcaster from your hand, then you get to destroy a monster on your opponents side of the field
A Fusion Summon is considered a special summon, so anything that can respond to a special summon (such as Bottomless Trap Hole) can be used against it.
It depends on the circumstances. Generally, "Bottomless Trap Hole" will be the better card; it works on Special Summons and it removes the monster it destroys from play. However, "Trap Hole" has one advantage: the monster summoned only needs 1000 ATK to trigger it, whereas "Bottomless Trap Hole" requires at least 1500 ATK. So, it would depend upon the type of deck you planned to use or went up against, but most people would choose "Bottomless Trap Hole".
"Botttomless Trap Hole" is typically viewed as being better than "Trap Hole" since the destroyed monster in banished/removed from play. A majority of cards function off of cards in the Graveyard or even activate in the Graveyard, so banishing the monster would be more disruptive. That being said, "Trap Hole" can destroy monsters with an ATK under 1500 unlike its "Bottomless" counterpart. Plus there are a few monsters like "D.D. Survivor" that function better when removed from play.
he wants to escape the consequences of his choices
Bottomless Trap Hole only removes the monster it destroys. If another Six Samurai destroyed itself to save the one BTH was used against, then neither the initial Six Samurai is removed from the game, nor the destroyed one, since neither were destroyed by Bottomless Trap Hole.
No
Only those that specifically say they destroy a card(s) on the field. Stardust can negate cards such as Mirror Force, Heavy Storm, Bottomless Trap Hole, Crush Card Virus (if it would destroy at least one monster), Mystical Space Typhoon, .... It can't negate cards such as Giant Trunade, Compulsory Evacuation Device, Dimensional Prison, D.D. Assailant, Reinforcements of the Army, ...