Is it correct that everything dry is pure even if it has impurity on it?
If something impure becomes dry, it remains impure and is not purified by drying. However, the impurity does not transfer by touching it if the one touching it is also dry. And Allah the Almighty knows best.
Can someone who begins a voluntary fast break it?
It is preferable for someone who begins an act of worship not to break it.
Allah the Exalted has said {what means}: "and let not your [good] deeds come to nought!" [Muhammad/33].
However, if a person starts a voluntary fast (nafl) and needs to break it, they are going against what is preferable, but there is no sin upon them.
Is it permissible to purchase the Udhiyah on installments?
It is permissible to purchase the sacrificial animal (uḍḥiyyah) on installments or by borrowing its price. However, it is not recommended for the poor to do so, because they are not required to offer a sacrifice, and Allah does not burden a soul beyond its capacity. If doing so would lead to negligence in providing for one's dependents (nafaqah wājibah), then giving precedence to obligatory maintenance over borrowing to buy the sacrifice is appropriate. This is based on the saying of the Prophet (peace and blessings be upon him): "It is sufficient sin for a person to withhold food from those whom he is responsible to support."
Furthermore, the sacrifice is valid from one who has a debt, but it is preferable (awlā) to repay the debt first, especially if the debt is due immediately (ḥāll). And Allah Almighty knows best.
For some time I have been searching for a guiding shaykh to teach me the spiritual path (*sulūk*) and the Islamic sciences, but I have not found one. What should I do?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
There is abundant good in this Ummah, and the scholars who guide and nurture are never cut off. The student of knowledge should strive in seeking them out, ask diligently after them, and draw from their learning.
As for one who cannot find a guiding and nurturing shaykh, he should send abundant prayers and peace upon our master the Prophet (peace and blessings be upon him), while keeping constantly to dhikr and worship and abstaining from the forbidden. For that will bring him to his goal in traveling toward Allah the Exalted. Imam Muḥammad al-Sanūsī (may Allah have mercy on him) said: “I have seen it said by some of the imams of Sufism that whoever lacks shaykhs of spiritual training (shuyūkh al-tarbiyah) should increase his invocation of blessings upon the Prophet (peace and blessings be upon him), for by it he will reach his goal.”
The scholar Aḥmad Zarrūq al-Mālikī (may Allah have mercy on him) said in Qawāʿid al-Taṣawwuf (p. 86), rule no. 114: “Dhikr is the decree of sainthood (manshūr al-wilāyah). Whoever is given dhikr has been given the decree. Our shaykh Abū al-ʿAbbās al-Ḥaḍramī (may Allah be pleased with him) said: ‘Keep constantly to dhikr and abundant blessings upon the Messenger of Allah (peace and blessings be upon him), for it is a ladder, an ascent, and a path to Allah the Exalted when the seeker does not meet a guiding shaykh.’” And Allah Almighty knows best.