QYour hBP01-081〈Hoshimachi Suisei〉 uses Arts 'Shining Comet' and downed center holomem 〈Amane Kanata〉 with hBP01-116〈Upao〉 attached.
While resolving the down, you use hBP01-007〈Hoshimachi Suisei〉's SP Oshi skill 'Shooting Star' to down opponent’s back holomem with hBP01-116〈Upao〉 attached.
When will each of the hBP01-116〈Upao〉's abilities resolve?

AThe order for resolution for this scenario follows the following order: first, any rule resolutions (e.g. Resolving downed holomem). Then, resolve all abilities that belong to the turn player that is pending resolution (e.g. Any abilities that has been activated and is pending resolution). Finally, the non-turn player will resolve all abilities that belong to the non-turn player that is pending resolution (e.g. abilities that trigger when holomem takes damage).

As hBP01-116〈Upao〉is an ability that belongs to the non-turn player, it will be resolved last.

QAn opponent’s hBP01-116〈Upao〉 is attached to 〈Amane Kanata〉 that you damaged, and you end your turn after your center holomem is downed and you have only collab holomem and no back holomem.
Does the next turn start with your center position empty?

AYes, the opponent's turn begins with you having only the collab holomem on stage without setting center holomem.

QIf your Life is 2 or more and your hBP01-014〈Amane Kanata〉 with hBP01-116〈Upao〉 attached is downed by opponent’s center holomem's Arts, can you deal damage to opponent’s center holomem due to hBP01-116〈Upao〉's ability?

AYes, damage is dealt.
hBP01-014 〈Amane Kanata） receiving damage triggers hBP01-116 〈Upao）'s 'when the holomem this mascot is attached to takes damage' ability, which goes into standby.

Then, rule-based down resolution archives hBP01-014 and hBP01-116, and resolves that standby effect of 'when the holomem this mascot is attached to takes damage' at the check timing.

QYour stage has no back holomem, only center holomem and collab holomem. Your center holomem uses Arts dealing damage to opponent holomem with hBP01-116〈Upao〉 attached, and your center holomem is down by its ability and archived.

What happens when there is no center holomem and back holomem?

AWithout setting center holomem, pass the turn to the opponent with only your collab holomem on stage.

QWhen your Life is 1 and your hBP01-013〈Amane Kanata〉 with hBP01-116〈Upao〉 attached is downed by opponent's Arts damage, can hBP01-116〈Upao〉's ability deal special damage to opponent holomem?

ANo, Damage is not dealt.
At HP 0, rule-based down resolution causes immediate loss without resolving hBP01-116 〈Upao）'s effect in standby.

QDoes hBP01-116〈Upao〉's mascot ability 'The holomem this mascot is attached to gets Arts+10.' also apply to special damage?

ANo, it does not apply.

QDoes hBP01-116〈Upao〉's ability 'when the holomem this mascot is attached to takes damage' trigger and apply to special damage?

AYes, it activates.

QIf hBP01-001〈Amane Kanata〉's Oshi skill 'Squish Squish' is used against an opponent center holomem with hBP01-116〈Upao〉 attached, will your center holomem receive special damage by hBP01-116〈Upao〉's ability?

ANo, it does not receive it.
hBP01-001 〈Amane Kanata）'s Oshi skill 'Squish Squish' is not a damage dealing ability, so hBP01-116 〈Upao）'s ability does not activate.