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].

QMy [Shirakami Fubuki] with hSD14-011[Kinda Yes-Ligator] attached downed my opponent's hSD17-008[Hoshimachi Suisei].
In this case, do we resolve the ability of hSD14-011[Kinda Yes-Ligator], or the ability of hSD17-008[Hoshimachi Suisei]'s gift "A Song for Tomorrow" first?

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

QMy [Shirakami Fubuki] with hSD14-011[Kinda Yes-Ligator] attached downed my opponent's hSD16-007[Sakura Miko].
In this case, do we resolve the ability of hSD14-011[Kinda Yes-Ligator], or the ability of hSD16-007[Sakura Miko]'s gift "Plop Your Head on Miko's Lap" first?

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

QMy [Shirakami Fubuki] with hSD14-011[Kinda Yes-Ligator] attached downed my opponent's hSD14-009[Shirakami Fubuki] with mascot attached.
In this case, do we resolve the ability of hSD14-011[Kinda Yes-Ligator], or the ability of hSD14-009[Shirakami Fubuki]'s gift "Can't Afford to Collapse" first?

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

QWhen my [Shirakami Fubuki] with hSD14-011[Kinda Yes-Ligator] attached downed my opponent's holomem, when do I send the top card of my cheer deck to my holomem?

AFor the down resolution of your opponent's holomem, the "if... downed" ability of [Shirakami Fubuki] with hSD14-011[Kinda Yes-Ligator] attached will be resolved first.
In this case, send the top card of your cheer deck to your holomem, your opponent archives their holomem, and reduces their life by 1. Your opponent turns it face up, and sends that cheer to a holomem on their stage.