What is the ruling on vomiting, and does it invalidate ablution?
Vomit is impure (najis). Its exit is not considered one of the nullifiers of ablution. However, the mouth must be washed and purified from it, and any that gets on clothing or the body must be washed for prayer, because prayer is not valid with impurity present on the body or clothing. And Allah the Almighty knows best.
Does Laylat al-Qadr move between the nights of Ramadan?
Laylat al-Qadr is definitely in Ramadan, and it is most likely in the last ten nights, especially on the odd-numbered nights.
For this reason, the Prophet ﷺ commanded observing worship in the last ten nights, and he himself used to dedicate those nights to worship. Many people put special effort into the 27th night of Ramadan, and they are rewarded in any case, whether they actually coincide with Laylat al-Qadr or not, because worshiping on any night of Ramadan carries great rewards.
What is the ruling of Islamic Law on leaving the Sunnah prayer of Dhuhr due to fatigue?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
According to the general rule, a Muslim should strive to maintain the Sunnah prayers before and after the obligatory prayers (al-sunan al-qabliyyah wa al-ba'diyyah). One is permitted to pray them sitting from the outset, but if one leaves them due to severe fatigue, there is no objection to doing so. It is recommended (mustahabb) to make up (qadā') a time-bound supererogatory prayer (nafl mu'aqqat) at any time if it was missed.
Al-Shirbīnī (may Allah have mercy on him) said: "If a time-bound supererogatory prayer is missed, it is recommended to make it up according to the more authoritative view (al-azhar), based on the hadith recorded in the two Ṣaḥīḥs: 'Whoever sleeps through a prayer or forgets it should pray it when he remembers it.' This is also supported by the fact that the Prophet (peace and blessings be upon him) made up the two rak'ahs of Fajr when he and his companions overslept in the valley and missed the dawn prayer until the sun had risen [narrated by Abū Dāwūd with an authentic chain]." And Allah, the Most High, knows best.
I bought an apartment, and I am saving money to pay the remaining balance of its price — is zakāh obligatory on this money?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
Zakāh is obligatory on all saved/hoarded wealth. Once the total wealth reaches the niṣāb (minimum threshold liable for zakah— equivalent to the value of 85 grams of pure 24-karat gold) and a full ḥawl (lunar year) has elapsed over it, zakāh becomes obligatory at a rate of (2.5%) — even if the wealth is earmarked for basic future needs, and even if the wealth's owner is in debt. Debt does not prevent the obligation of zakāh, nor is it deducted when calculating the zakāh due, based on the general and unrestricted wording of all the Qur'anic verses commanding zakāh, which draw no distinction between someone in debt and anyone else.
It is stated in ‘Umdat as-Sālik, from the books of the Shāfi‘ī school: "If someone possesses exactly the niṣāb, and owes an equivalent amount in debt, zakāh remains obligatory on what is in his possession, and the debt does not prevent this obligation" — meaning, the obligation of zakāh on what is presently in his possession.
The ḥawl (annual cycle) recognized for zakāh, as well as for the other sharī‘ah obligations that Allah, the Exalted, has imposed upon Muslims, is the Hijrī (lunar) year. And Allah, the Exalted, knows best.
