Can you activate Mystical Space Typhoon?

Mystical Space Typhoon is a Quick-Play Spell card. You can activate it during your opponent's turn if you set it face-down in your turn. You can activate it during the Main Phase 1 & 2 in your turn.

Can MST destroy itself?

In general, if you cannot apply the card's effect when activating a card, then you cannot activate it. Thus, if there are no cards on the field that can be destroyed, then you cannot activate "Mystical Space Typhoon" nor a Set "Raigeki Break". Also, when you activate "Mystical Space Typhoon", it cannot target itself.

Does trap hole negate summon effects?

"Bottomless Trap Hole" does not negate a Summon. It destroys and removes from play [sic] the monsters after it has already been successfully Summoned.

Can you activate a trap card when it is destroyed?

Destroying trap/spell cards. If you can chain the trap/spell cards to the card that destroys them then yes, you can activate them before they are destroyed and the effect will still resolve if the effect doesn't require them to be on the field, if you can't chain them you're out of luck.