QYou archived two holo Power cards for hBP01-006〈Takanashi Kiara〉's Oshi skill 'Phoenix Tail' cost. If there is a holomem from among the two cards, can you return the archived holomem to hand with 'Return 1 holomem from your archive to hand' effect?

AYes, it can be returned.
If no holomem is in archive and you archive 2 holo Powers as cost but no holomem appear, do nothing and finish the ability resolution.

QIf your Buzz holomem hBP01-071〈Omaru Polka〉 is downed, and you use hBP01-006〈Takanashi Kiara〉's SP Oshi skill 'Rise from the ashes', how much life damage do you receive?

AhBP01-071 〈Omaru Polka）'s Extra ability is a continuous replacement effect, so life damage when downed is 2.
Applying hBP01-006 〈Takanashi Kiara）'s SP Oshi skill 'Rise from the ashes' reduces life damage by 1, making life damage 1.

QCan hBP01-006〈Takanashi Kiara〉's Oshi skill 'Phoenix Tail' be used even if there are no holomem in the archive?

AYes, it can be used.
Abilities of holomem or Oshi holomem can be used even if the specified cards cannot be chosen.

QWhen hBP01-006〈Takanashi Kiara〉's SP Oshi skill 'Rise from the Ashes' is used, do you add any cheers attached to the holomem returned to hand along with the holomem themselves?

ANo, only holomem are added to hand.
Cheers and support cards attached to holomem are archived as usual.