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.

QYour opponent's Oshi holomem is hBP01-004〈Usada Pekora〉 and you reduced opponent's center holomem HP to 0 with hBP01-081〈Hoshimachi Suisei〉's Arts ‘Shining Comet’. If both players want to use SP Oshi skills ‘Shooting Star’ and ‘Little Wild Rabbits~’ respectively, in which order will they be resolved?

ADown is resolved when the holomem HP reaches 0, and once the down is confirmed, the Oshi skill 'Little Wild Rabbits~' can be used.
After resolving the effect, send the cheer from Life to 1 holomem, then use the SP Oshi skill 'Shooting Star'.

QYou brought opponent's center holomem HP to 0 with hBP01-081〈Hoshimachi Suisei〉's Arts 'Shining Comet'. If you use your SP Oshi skill 'Shooting Star', in which order should opponent sending cheers from life and your SP Oshi skill be resolved?

ADown is resolved when the holomem HP reaches 0, so first send the cheer from life to holomem, then use the SP Oshi skill 'Shooting Star'.

QFor hBP01-081〈Hoshimachi Suisei〉's Arts 'Shining Comet', if you do not 'archive 2 ▲blue▲ cheers from this holomem:' as stated, does the ability apply to opponent back holomem?

ANo, a back holomem cannot be targeted by Arts.
If the cost is not resolved, the effect is not resolved, and Arts damage can only target the center holomem or collab holomem.

QIf your hBP01-081〈Hoshimachi Suisei〉 is stacked as the third card (above debut and 1st holomem) and uses Arts 'Shining Comet' while archiving 2 blue cheers, is the damage dealt 180 or 240?

AIt becomes 180 damage.
Since 2 cards (1st holomem and hBP01-081 〈Hoshimachi Suisei）) are stacked on debut holomem, +120 is added to 60 Arts damage.