QMy [Shirakami Fubuki] with hSD14-011[Kinda Yes-Ligator] attached downed my opponent's hSD18-007[Mori Calliope].
In this case, do we resolve the ability of hSD14-011[Kinda Yes-Ligator], or the ability of hSD18-007[Mori Calliope]'s gift "A Small but Special Present" first?

AThe turn player's abilities will be resolved first, so start resolution with the ability of hSD14-011[Kinda Yes-Ligator].

QWhen my hSD18-007[Mori Calliope] is downed during my opponent's turn, when do I archive the top card of my deck with the ability of my gift "A Small but Special Present"?

AFor the down resolution of hSD18-007[Mori Calliope], the "if... downed" ability will be resolved first.
In this case, archive the top card of your deck, archive hSD18-007[Mori Calliope], and reduce your life by 1. Turn it face up, and send that cheer to a holomem on your stage.

QMy hSD15-008[Juufuutei Raden] downed my opponent's hSD18-007[Mori Calliope].
In this case, do we resolve the ability of hSD15-008[Juufuutei Raden]'s gift "Twilight", or the ability of hSD18-007[Mori Calliope]'s gift "A Small but Special Present" first?

AThe turn player's abilities will be resolved first, so start resolution with the ability of hSD15-008[Juufuutei Raden].