If “Allure Queen LV5” is flipped face-down after being Special Summoned by the effect of “Allure Queen LV3”, it is no longer treated as having been Special Summoned by “Allure Queen LV3”. So you can no longer use its first effect to equip it with a monster. (However, it is still considered to have been Special Summoned for effects like “Special Hurricane”.)
If “Allure Queen LV5” is Special Summoned by the effect of “Allure Queen LV3”, and is then temporarily removed from play with “Dimensionhole” or “Interdimensional Matter Transporter”, after it returns to the field it can still use its first effect. The fact that it was Special Summoned by “Allure Queen LV3” is not reset.
If “Allure Queen LV5” is Special Summoned by the effect of “Allure Queen LV3”, and is then permanently removed from play with “D.D. Warrior”, if it is later Special Summoned by an effect like “Return from the Different Dimension”, it is no longer treated as being Special Summoned by “Allure Queen LV3”. So it cannot use its first effect anymore.