Is the Tarawih prayer valid if the sister leads it despite her numerous mistakes in reciting the the Qur’an, and her refusal to be led?

If your sister does not distort the meaning by mispronunciation (laḥn) in Al-Fatiha, and does not intentionally mispronounce in other parts, then it is valid to follow her, and your prayer behind her is permissible, though disliked. This is because it is disliked for someone who frequently mispronounces to lead the prayer, and it is more befitting that one who does not mispronounce leads you. So, your prayer behind your sister is valid, but it is preferable that you lead her, or that someone who does not mispronounce frequently leads you in prayer.