The effect he uses to destroy a monster, is an Ignition Effect, so can only be used in your own turns, in your main phases. However his counter-adding effect will add counters in either turn.
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No. It is an Ignition Effect, meaning you may only activate Snipe Hunter's effect during your turn.
Yes, it is a Quick Effect and can be activated in either player's turn.
No, Alector has no effects that can be used in the opponent's turn. His summon effect is something that can only be done in your main phases, and his negation effect is an Ignition Effect so again, can only be done in your turn.
As Destiny Hero - Plasma's equip effect is a "Once Per Turn" effect, yes it can, and the existing monster is destroyed.
You can only do so if a card effect specifically lets you, like Call of the Haunted. Otherwise, regular Special Summons, Synchro, Fusions, XYZ's, etc, all must be done on your turn in your own main phases.