QIf hBP01-121〈Kotori〉 is attached to your 1st holomem hBP01-065〈Takanashi Kiara〉 and you Bloom with a 1st holomem hBP01-066〈Takanashi Kiara〉, can you draw a card with hBP01-121〈Kotori〉's ability?

ANo, you cannot.
Bloom level up refers to blooming from debut to 1st holomem, or 1st holomem to 2nd holomem. Blooming into same level is not considered level up.

QWhich card can be archived when using hBP01-066〈Takanashi Kiara〉's Arts 'Kneel Before Me.' ability to archive 1 holomem overlapping this holomem?

AYou can only archive holomem cards stacked under this card, not this card itself.
For example, if the bloomed holomem has 1 each of debut holomem and 1st holomem stacked, this ability can be activated only once.

QCan you archive this card for hBP01-066〈Takanashi Kiara〉's Arts 'Kneel Before Me.' ability 'used to archive a holomem attached to this holomem'?

ANo, it cannot.

QIf your opponent uses hBP01-005〈Takane Lui〉's SP Oshi skill 'Hawkeye' on your hBP01-066〈Takanashi Kiara〉, can your hBP01-066〈Takanashi Kiara〉 use 'Kneel Before Me.'?

AYes, it can be used.
It only restricts movement of center holomem and back holomem, so cards stacked on holomem are unaffected by that ability.