Creature abilities can be played as instants, so they can be used on the opponent's turn.
However, creatures with summoning sickness cannot use their tap abilities until the player's next turn.
During the opponent's turn, the most significant tactical aspects are the untap step, the upkeep step, the first main phase, the beginning of combat step, and the second main phase.
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First question, your creatures have summoning sickness until your next turn. That means you can't use the tap ability until your next turn, not your opponents next turn.
If you cast a spell to destroy an artifact, the opponent can respond to you casting the card, and activate the ability of the urn.
When can I use a tap ability? : r/CasualMTG - reddit