1. Correct, their only ability doesn't produce mana, so Blood Sun makes them lose it. They end up having no ability of any kind.
2. No, they enter untapped, because the rules tell us to look at what the permanent would look like on the battlefield to know if effects that change how the permanent enters the battlefield should apply. Since the Village wouldn't have the ability on the battlefield, it doesn't apply.
3. Same as 2. It enters untapped and you don't get the option to pay 2 life, that ability can't apply because the land wouldn't have it on the battlefield.
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2. Lands like treetop village. Do they still come into play tapped if a Blood Sun is in play?
3. Do shock lands like Stomping Grounds still come into play tapped if a Blood Sun is in play?
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2. No, they enter untapped, because the rules tell us to look at what the permanent would look like on the battlefield to know if effects that change how the permanent enters the battlefield should apply. Since the Village wouldn't have the ability on the battlefield, it doesn't apply.
3. Same as 2. It enters untapped and you don't get the option to pay 2 life, that ability can't apply because the land wouldn't have it on the battlefield.
+1
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Want to play a UW control deck in modern, but don't have jace or snaps?
Please come visit us at the Emeria Titan control thread