QI used hBP03-006[Inugami Korone]'s Oshi skill "Infinite Stamina", but since there are no holomem at rest nothing happens. In this case, can I deal 80 damage with hBP06-069[Inugami Korone]'s Arts "Hard Beating"?

ANo, as the holomem did not become active from the Oshi skill "Infinite Stamina".

QI used hBP03-006[Inugami Korone]'s Oshi skill "Infinite Stamina", but since there are no holomem at rest nothing happens.
In this case, can I deal 210 damage with hBP06-070[Inugami Korone]'s Arts "Ragnarok WowWowAxe"?

AYes, you can.

QMy [Inugami Korone] at rest becomes active with the ability of my hBP03-006[Inugami Korone]'s Oshi skill "Infinite Stamina".
Later in the turn, if I bloom to hBP06-069[Inugami Korone] and use Arts "Hard Beating", will I get +50?

AYes, you will.

QWhen I use both the gift "Wan-Dafuuul~♡" of hBP03-066〈Inugami Korone〉 and the SP Oshi skill "Wow Wow Wow Wow" of hBP03-006〈Inugami Korone〉, which ability should I resolve first?

AAs both abilities trigger "when downed", you may resolve them in any order.

QMy hBP03-066〈Inugami Korone〉 with 3 cards stacked (hBP03-066〈Inugami Korone〉 stacked on a Debut holomem and a 1st holomem) was downed during the opponent's turn.
While I would like to return holomem using the SP Oshi skill "Wow Wow Wow Wow" of hBP03-006〈Inugami Korone〉, which card does "1 card from among the stacked holomem including that downed holomem" refer to?

AIn this case, as hBP03-066〈Inugami Korone〉 is stacked on a Debut holomem and a 1st holomem, you can return any 1 card from among the 3 cards to hand.

QWhen 2 of my 〈Inugami Korone〉 were downed by the ability of Arts "Rain Mantra" of the opponent's hBP01-087〈Kobo Kanaeru〉, can I use the SP Oshi skill "Wow Wow Wow Wow" of hBP03-006〈Inugami Korone〉 to reattach the cheers from both holomem and return holomem to my hand?

ANo. As down processing resolves for 1 holomem at a time, you may only use it for either 1 of the downed holomem.