The current game-legal text for Slifer the Sky Dragon says "When Normal Summoned, cards or effects cannot be activated" This prevents the activation of Bottomless Trap Hole in response to Slifer's Normal summon.
However this is only for Normal Summon. Slifer can be Special Summoned, and if Special Summoned, then his continuous effect will apply as normal, before cards that respond to summons can be activated. That means if he is special summoned, and his ATK goes to 2000, the summon can be responded to by Bottomless Trap Hole, he will be destroyed and removed.
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Bottomless Trap Hole does not target, so Tyrant Dragon is unable to negate it.
According to the effects of Slifer the Sky Dragon given in Yu-Gi-Oh! 5D's World Championship 2010: Reverse of Arcadia, Bottomless Trap Hole may not be activated when it is Tribute Summoned onto the field. It clearly states that "...when this card is Normal Summoned, the effects of Spell, Trap, and Effect Monster Cards cannot be activated..."Slifer the Sky Dragon can be destroyed afterwards by cards that do not target it, such as Mirror Force, or Widespread Ruin.
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".
yes You can
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.