Is Ṭahārah (Purification) Required for Ṭawāf?

The opinion that it is permissible [to perform ṭawāf without wuḍū’] is stronger, but a person should not do this unless there is a need for it, such as breaking the wuḍū’ during times of congestion – then the need for this opinion is required – because a person is unable to leave and perform wuḍū’. Or perhaps a person has companions, and if he leaves, he will delay them. But if a person is not busied or under any preventative measures, then they should not perform ṭawāf except while being in the state of having wuḍū’ – so as to avoid scholarly disagreement. “Leave what causes you doubt for that which does not cause you doubt”.

And this opinion that we stated, it was the choice [opinion] of Shaykh al-Islām, Ibn Taymiyyah, may Allāh have mercy upon him.

What is the ruling on the removal of hair and clipping the nails for the one who wants to perform an uḍḥiyah but is not performing the Ḥajj?

Conclusion: The ḥadīth of Umm Salamah restricting one from removing from their hair or nails until their ūḍḥiyah has been slaughtered is confirmed to be from her words; as for the marfū’ narrations of this ḥadīth, then they appear to be incorrect. The ḥadīth of Umm Salamah appears to also be specific to those who are performing the Ḥajj and in the state of iḥrām. As for those who are not performing the Ḥajj, then they are not restricted from doing what those who are in the state of iḥrām are prohibited from doing.

And Allāh (ﷻ) knows best.

Response to the claim: there is ijmā’ that if a person intentionally vomits then they have nullified their fast & must make up the day

There is nothing authentic to prove that vomiting nullifies one’s fast; whether one vomits intentionally or unintentionally. As for the Ḥadīth: “And whoever makes himself vomit then he must makeup (the fast).”, then the correct opinion is that it is weak. It was weakened by Aḥmad, Al-Bukhārī, At-Tirmiṫhī and An-Nasāʿī. The reason for the weakness is ʿĪsā Ibn Yūnus narrated the Ḥadīth in a mistaken manner. Likewise, Abū Hurayrah was of the opinion that the fast isn’t nullified by anything coming out of the body.