What type of illness that permits breaking the fast in Ramadan?
It is an illness that, if one fasts, there is a fear it may lead to their death or cause unbearable hardship.
What is the ruling of Islamic Law on eating or drinking during the circumambulation?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
It is disliked (makrūh) to eat or drink during the circumambulation (ṭawāf).
Shaykh al-Islām Imām al-Nawawī states in al-Majmūʿ: "It is disliked to eat or drink during ṭawāf, and the dislikedness of drinking is lighter. Neither of them, nor both together, invalidates the ṭawāf. Al-Shāfiʿī said: 'There is no objection to drinking water during ṭawāf, and I do not consider it sinful; however, I prefer that it be avoided, as refraining from it is more befitting in terms of proper conduct.' Among those who explicitly stated the dislikedness of eating and drinking, and that drinking is the lesser of the two, is the author of al-Ḥāwī."
If, however, a person is in genuine need of drinking, then there is no dislikedness in doing so. In any case, the ṭawāf itself remains valid. And Allah the Almighty knows best.
Is washing the private part after urinating (Istinjaa`) a condition, and is it done with water and soap, or with water only?
Istinjaa` is obligatory for removing impurity, and it can be done with toilet paper, or a stone, or water. It is preferable to do Istinjaa` first with paper, or stone, or the like, then to wash the private part with water until making sure that the impurity has been removed. Using soap is not a condition, but there is no harm in doing so provided that it is washed off with water. It is also permissible to use either water, or paper if impurity is removed by any.
Does the Obligatory Ritual Bath (Ghusl) Suffice in Place of Ablution (Wuḍūʾ)?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
The more prudent course is to perform ablution (wuḍūʾ) after the ritual bath (ghusl), so as to exit the difference of scholarly opinion and because the sufficiency of the ritual bath in place of ablution is contingent upon a number of conditions and requirements being met. And Allah the Almighty knows best.