Can you use Mystical Space Typhoon on your opponent turn?

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 cannot activate Quick-Play Spell cards the turn they are set.

Can counter traps be activated the same turn?

Before you can activate a Trap Card, you must Set it on the field first. You cannot activate a Trap in the same turn that you Set it, but you can activate it at any time after that—starting from the beginning of the next turn.

Does Mystical Space Typhoon negate effects?

"Mystical Space Typhoon" and "Dust Tornado" do not negate the effects of cards they destroy. So Spell and Trap Cards that "Mystical Space Typhoon" is chained to may still resolve if they are in the Graveyard.

Can Mystical Space Typhoon negate dark hole?

It does not negate, unless it's a continuous. Since continuous effects need to be activated while it's face up. If the continuous card were to be destroyed while being activated, it just fails. Just to clarify, mst does not negate continuous effects.