QIf hBP01-034〈Aki Rosenthal〉 with hBP01-114〈Stone Axe〉 attached restores due to hBP01-037〈Aki Rosenthal〉's bloom effect 'Gothic Doll', can you draw cards via hBP01-114〈Stone Axe〉's ability?

AYes, it can be drawn.

QIf hBP01-034〈Aki Rosenthal〉 has both hBP01-114〈Stone Axe〉 and hBP01-119〈Jobs〉 attached and 10 HP, can you deal Arts damage to opponent holomem?

AYes, you can deal damage.
After declaring Arts usage, hBP01-114 〈Stone Axe） and hBP01-119 〈Jobs）'s 'When Arts is used' abilities simultaneously trigger and become standby.
After dealing Arts damage, you choose the resolution order. If you activate 〈Jobs） first to restore 10 HP, then apply 〈Stone Axe） dealing 10 damage, hBP01-034 〈Aki Rosenthal） does not become downed.

QIf hBP01-034〈Aki Rosenthal〉 with 10 HP and hBP01-114〈Stone Axe〉 attached uses Arts, can you deal Arts damage to opponent holomem?

AYes, you can deal damage.
When hBP01-034 〈Aki Rosenthal） with remaining HP 10 uses Arts while equipped with 〈Stone Axe）, after dealing Arts damage, 〈Stone Axe）'s 'When Arts is used' ability activates, causing 〈Aki Rosenthal） to receive 10 damage and be downed.

QhBP01-034〈Aki Rosenthal〉 with 10 HP and hBP01-114〈Stone Axe〉 attached uses Arts, gets damaged by hBP01-114〈Stone Axe〉 ability, and is downed. If the center holomem position becomes vacant, does the center holomem remain vacant until your next reset phase?

ANo, the center holomem will be set at the end phase of that turn.

If any back position holomem that is not resting exist, choose one from among them and move it to the center position.

If only resting holomem exist, move that holomem to center position in resting position.

QIf hBP01-034〈Aki Rosenthal〉 with 10 HP and hBP01-114〈Stone Axe〉 attached uses Arts and goes down, does your life decrease?

AYes, the life decreases.
Even if your own holomem is dealt Damage and downed by your card's ability, if there is no special notation regarding life increase or decrease, Life decreases.