Source: https://www.db.yugioh-card.com/yugiohdb/faq_search.action?ope=4&cid=12441&request_locale=ja
①: If this card is Normal or Special Summoned: You can target 1 face-up monster your opponent controls; change it to face-down Defense Position.
②: If this card is in your GY (Quick Effect): You can banish 7 other cards from your hand, field, and/or GY; Special Summon this card.
①: "If this card is Normal or Special Summoned: You can target 1 face-up monster your opponent controls; change it to face-down Defense Position." is a Trigger Effect. (It targets 1 face-up monster the opponent controls, also it can be activated during the Damage Step.)
②: Even if "Fairy Tail - Snow" is Special Summoned by "Ultimate Offering" or "Call of the Haunted" at Chain Link 2 or higher, you can activate the monster effect, "You can target 1 face-up monster your opponent controls; change it to face-down Defense Position."
Side NOTE: (BTW, the effect would activate in a new chain.)
③: "If this card is in your GY (Quick Effect): You can banish 7 other cards from your hand, field, and/or GY; Special Summon this card." is a Quick Effect that activates in the Graveyard. The effect can be activated during the opponent's turn. (It does not target and it cannot be activated during the Damage Step.)
④: When you activate the monster effect, "Special Summon this card.", banishing 7 cards from the GY is a cost. (You cannot banish the "Fairy Tail - Snow" that activates its effect.)
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Question:
While the effect of Necrovalley is applying, if I activate the effect of a Fairy Tail – Snow in my Graveyard, can I banish cards from my Graveyard along with Necrovalley as the activation cost?
Answer:
In this scenario, while the effect of Necrovalley is applying, you cannot banish cards from your Graveyard, so you cannot activate the effect of a Fairy Tail – Snow in your Graveyard unless you can banish 7 cards from your hand and/or field. (Even if you banish Necrovalley among the 7 cards, you cannot banish any cards from your Graveyard.) Also note that the effect of Fairy Tail – Snow is an effect that is applied to a card(s) in the Graveyard, so if Necrovalley is not banished as part of the cost to activate the effect, the effect of Fairy Tail – Snow will be negated.
Source: http://www.db.yugioh-card.com/yugiohdb/faq_search.action?ope=5&fid=18770&keyword=&tag=-1
YGORG: https://ygorganization.com/ifonlyrickgrimeshadupdatednecrovalleyssozombieswouldntcomeback/
Inquiry Insertion Date: 03.24.17 (Moved from an FAQ into a Database Inquiry)
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Since I could not find a Q&A translated on YGORG for the two remaining scenarios in the database, I've simplified them to a short response, excluding the questions. However, you still have access the original database citation.
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The Quick Effect of Fairy Tail - Snow can be activated multiple times in the same chain.
Source: https://www.db.yugioh-card.com/yugiohdb/faq_search.action?ope=5&fid=18665&keyword=&tag=-1
Inquiry Insertion Date: 03.24.17 (Moved from an FAQ into a Database Inquiry)
The Quick Effect of "Fairy Tail - Snow" cannot be activated while the effect of "Vanity's Emptiness" is applying.
Source: https://www.db.yugioh-card.com/yugiohdb/faq_search.action?ope=5&fid=18772&keyword=&tag=-1
Inquiry Insertion Date: 03.24.17 (Moved from an FAQ into a Database Inquiry)